text:
\part{Theory}
\chapter{General Theory} \label{genteoria}
\section{Introduction}
The object of study of the present investigation is the fundamental
equation of classical hydrodynamics of an \textit{ideal,
incompressible, viscous fluid} subject to some external forces,
namely the Navier Stokes equation in three dimensional Euclidian
space $\mathbb{R}^3$, which, in our adopted notation, reads as
follows:
\begin{equation}\label{EulerusEqua}
    \frac{\partial}{\partial t}\mathbf{u}\, + \, \mathbf{u}\cdot \nabla \mathbf{u} \, =
     \, - \, \nabla p  \, + \, \nu \, \Delta \,\mathbf{u} \, + \, \mathbf{f} \quad ;
     \quad \nabla \cdot \mathbf{u} \, = \, 0
\end{equation}
In equation (\ref{EulerusEqua}), $\mathbf{u} \, = \,
\mathbf{u}\left( x \, , \,t\right)$ denotes the local velocity
field, $p(\mathbf{x},t)$ denotes the local pressure field, $\nu$ is
viscosity and $\mathbf{f}$ is the external force, if it is
introduced. The symbol $\Delta$ stands for the standard laplacian:
\begin{equation}\label{laplaccio}
    \Delta \, = \, \sum_{i=1}^3 \,\frac{\partial^2}{\partial x_i^2}
\end{equation}
In vector notation eq. (\ref{EulerusEqua}) takes the following form:
\begin{equation}\label{EulerusEquaVec}
    \frac{\partial}{\partial t} u^i\, + \, u^j \, \partial_j \, u^i
    \, = \, - \, \partial^i \, p  \, + \, \nu \,\Delta u^i\, + \, f^i \quad ;
    \quad \partial^\ell  \, u_\ell\, = \, 0
\end{equation} and admits some straightforward rewriting that, notwithstanding
the kinder-garden arithmetic involved in its derivation, is at the
basis of several profound and momentous theoretical  developments
which have kept  the community of  dynamical system theorists busy
for already fifty years
\cite{arnoldus,Childress0,Childress,Henon,Dombre,ArnoldBook,Bogoyav0,Bogoyav,
arnoldorussopapero,FFMF,Dynamo,Gilbert,Etnyre2000,Ghrist2007}.
\par
In  the present paper we aim at extending to the case where $\nu
\neq 0$ previous results applying to the case of null viscosity,
namely to Euler equation. The scope, however is more ample since we
introduce a more direct reference to \textit{direct reference} to
contact structures and to the recent developments occurred in this
field of mathematics, where the notion of \textit{singular contact
structures} has been introduced
\cite{cardone2019,miranda2021,trasversozero,unochiuseinteg,guglielminoconmiranda,pollosingolare}
to account in particular for boundaries of a certain type
(cylidrical ends) which are potentially momentous for applications.
\par
The framework of our paper is the same of the previous paper
\cite{Fre:2015mla} which introduced into the classical field of
mathematical fluid-mechanics a new group-theoretical approach
allowing for a more systematic classification and an algorithmic
construction of the so named Beltrami flows, hopefully providing new
insight into their properties. Combining the group theoretical
classification of Beltrami fields and their generalized relation
with \textit{contact structures} possibly \textit{admitting
singularities} is the most promising follow up of our work. Indeed
here we put on a more clear cut basis the group theoretical
foundations of the results of \cite{Fre:2015mla}, we extend the in
depth analysis of the cubic lattice case there provided to that of
the hexagonal lattice, finally showing that, in this way, we have
covered all possible cases and symmetry groups. Last but not least
we provide an user's guide to a rather complete set of MATHEMATICA
Codes that allow for the explicit construction of a large amount of
NSE exact periodic solutions with an efficient mastering of their
symmetries and the options for plotting the associated streamlines.
\par
The main new conceptual item contributed by \cite{Fre:2015mla} is
the notion of \textbf{Universal Classifying Group} that is an
intrinsic property of the considered crystallographic lattice
$\Lambda$ and of its point group $\mathfrak{P}^{max}_\Lambda$,
which, by definition, is the maximal finite subgroup of
$\mathrm{SO(3)}$ leaving the lattice $\Lambda$ invariant.
\par
The reason why lattices and crystallography are brought into the
study of fluid dynamics is that  we focus on \textit{hydro-flows}
that are confined within some compact space, as it happens in a
large variety of cases of interest for technological applications
like industrial autoclavs, pipelines, thermal machines of various
kind, blood vessels in physiology, liquid helium micro-tubes in
superconducting magnets, chemical reactors with mechanical agitation
sytems and so on. The argument goes as follows. Solutions of partial
differential equations (PDE.s) like the NS-equation in
(\ref{EulerusEqua}), that encode the characterizing feature of being
confined to finite regions of space can be obtained essentially by
means of two alternative strategies:
\begin{description}
  \item[A)] By brutally imposing boundary conditions that simulate the
  walls of the chamber, tube, box or whatever else contains the
  flowing fluid. This strategy is the most direct and suitable for
  numerical computer aided integration of the PDE.s but it is hardly
  viable in the search of exact analytic solutions of the same PDE.s with the
  ambition of establishing some rational taxonomy.
  \item[B)] The use of periodic boundary conditions which amounts to
  restricting one's attention to a compact space $\mathcal{M}_3$ without boundary ($\partial \mathcal{M}_3 \, = \, 0$) as
  a mathematical model of the finite volume region of interest.
\end{description}
The use of alternative B) is very much convenient from the
mathematical viewpoint and it is much less restrictive than one
might think a priori. Indeed it corresponds to the higher
dimensional generalization of the well known possibility of
developing in Fourier series any compact support function $f(x)$
irrespectively of its periodical or non periodical global behavior
outside of the finite interval where it is represented as the sum of
a Fourier series.
\begin{description}
  \item[C)] The already mentioned recent developments in contact geometry
\cite{cardone2019,miranda2021,trasversozero,unochiuseinteg,guglielminoconmiranda,pollosingolare}
that have established a systematic  way, within the formalism of
$\mathfrak{b}^m$-forms, of adding boundaries that correspond to
cylindrical ends to a compact manifold, in particular to a
$\mathrm{T^3}$ torus, open exciting perspectives of conjugating the
group theoretical classification of Beltrami fields with this new
vision.
\end{description}
This being clarified a systematic way of imposing periodic boundary
conditions  is the the identification of the $\mathcal{M}_3$
manifold with a $\mathrm{T^3}$ torus obtained by modding
$\mathbb{R}^3$ with respect to a three dimensional lattice $\Lambda
\subset \mathbb{R}^3$:
\begin{equation}\label{metricT3}
 \mathcal{M}_3 \, = \,    \mathrm{T}^3_g \, = \, \frac{\mathbb{R}^3}{\Lambda}
\end{equation}
Abstractly the  lattice $\Lambda$ is a an abelian infinite group
isomorphic to $\mathbb{Z}\times \mathbb{Z} \times \mathbb{Z}$ which
is embedded in some way into $\mathbb{R}^3$. Using
eq.(\ref{metricT3}) the topological torus
\begin{equation}\label{tritorustop}
    \mathrm{T^3} \simeq \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1
\end{equation}
comes out automatically equipped with a flat constant metric.
Indeed, according with (\ref{metricT3}) the flat Riemaniann space
$\mathrm{T}^3_g$ is defined as the set of equivalence classes with
respect to the following equivalence relation:
\begin{equation}\label{equivalenza}
 {\mathbf{r}}^\prime \, \sim \,  {\mathbf{r}} \quad \mbox{iff}
\quad  {\mathbf{r}}^\prime \, - \,  {\mathbf{r}} \, \in \, \Lambda
\end{equation}
The metric $g$ defined on $\mathbb{R}^3$ is inherited by the
quotient space and therefore it endows  the topological torus
(\ref{tritorustop}) with a flat Riemaniann structure. Seen from
another point of view the space of flat metrics on  $\mathrm{T}^3$
is just the coset manifold $\mathrm{SL(3,\mathbb{R})}/\mathrm{O(3)}$
encoding all possible symmetric matrices, alternatively all possible
space lattices, each lattice being spanned by an arbitrary triplet
of basis vectors.
\paragraph{\bf Lattices} To make the above statement
precise let us consider the standard $\mathbb{R}^3$ manifold and
introduce a basis of three linearly independent 3-vectors that are
not necessarily orthogonal to each other and of equal length:
\begin{equation}\label{sospirone}
 {\mathbf{w}}_\mu \, \in \, \mathbb{R}^3 \quad \mu \, = \, 1,
\dots ,\, 3
\end{equation}
Any vector in $\mathbb{R}$ can be decomposed along such a basis and
we have:
\begin{equation}\label{xvec}
 {\mathbf{r}} \, = \,  r^\mu {\mathbf{w}}_\mu
\end{equation}
The flat, constant metric on $\mathbb{R}^3$ is defined by:
\begin{equation}\label{gmunu}
g_{\mu\nu} \, = \, \langle   {\mathbf{w}}_\mu \, , \,
 {\mathbf{w}}_\nu \rangle
\end{equation}
where $\langle \, ,\, \rangle$ denotes the standard euclidian scalar
product. The space lattice $\Lambda$ consistent with the metric
(\ref{gmunu}) is the free abelian group (with respect to the sum)
generated by the three basis vectors (\ref{sospirone}), namely:
\begin{equation}\label{reticoloLa}
\mathbb{R}^3 \, \ni \,    {\mathbf{q}}  \, \in \, \Lambda \,
\Leftrightarrow \,  {\mathbf{q}} \, = \, q^\mu \,
 {\mathbf{w}}_\mu \quad \mbox{where} \quad q^\mu \, \in \,
\mathbb{Z}
\end{equation}
\paragraph{\bf Dual lattices} Any time we are given a lattice in the sense of the definition (\ref{reticoloLa}) we obtain
a dual lattice $\Lambda^\star$ defined by the property:
\begin{equation}\label{reticoloLastar}
\mathbb{R}^3 \, \ni \,    {\mathbf{p}}  \, \in \, \Lambda^\star \,
\Leftrightarrow \, \langle  {\mathbf{p}} \, , \,
 {\mathbf{q}}\rangle \, \in \, \mathbb{Z} \quad \forall \,
 {\mathbf{q}}\, \in \, \Lambda
\end{equation}
A basis for the dual lattice is provided by a set of three
\textit{dual vectors} $ {\mathbf{e}}^\mu$ defined by the
relations\footnote{In the sequel for the scalar product of two
vectors we utilize also the equivalent shorter notation $
{\mathbf{a}}\, \cdot  {\mathbf{b}} \, = \, \langle
 {\mathbf{a}}\, \cdot  {\mathbf{b}}\rangle $}:
\begin{equation}\label{dualvecti}
    \langle  {\mathbf{w}}_\mu \, , \,  {\mathbf{e}}^\nu \rangle \, = \, \delta^\nu_\mu
\end{equation}
so that
\begin{equation}\label{pcompi}
\forall \,  {\mathbf{p}} \, \in \, \Lambda^\star \quad
 {\mathbf{p}} \, = \, p_\mu \,  {\mathbf{e}}^\mu \quad
\mbox{where } \quad p_\mu \, \in \, \mathbb{Z}
\end{equation}
According with such a definition it immediately follows that the
original lattice is always a subgroup of the dual lattice and
necessarily a normal one, due to the abelian character of both the
larger and smaller group:
\begin{equation}\label{tarragonese}
    \Lambda \, \subset \, \Lambda^\star
\end{equation}
\section{Rewriting of the equations of  hydrodynamics in a
geometrical set up} Let us then begin with the rewriting of
eq.(\ref{EulerusEquaVec}) which is the starting point of the entire
adventure. The first step to be taken in our raising conceptual
ladder is that of promoting the fluid trajectories, defined as the
solutions of the following first order differential
system\footnote{In mathematical hydrodynamics people distinguish two
notions, that of trajectories, which are the solutions of the
differential equations (\ref{streamlines}) and that of streamlines.
Streamlines are the instantaneous curves that at any time $t=t_0$
admit the velocity field $u^{i}(x,t_0)$ as tangent vector.
Introducing a new parameter $\tau$, streamlines at time $t_0$, are
the solutions of the differential system:
\begin{equation}\label{strimotti}
    \frac{d}{d\tau}x^i(\tau) \, = \, u^i(\mathbf{x}(\tau),t_0)
\end{equation}
In the case of steady flows where the velocity field is independent
from time, trajectories and streamlines coincide.}:
\begin{equation}\label{streamlines}
    \frac{d}{dt}x^i(t) \, = \, u^i(x(t),t)
\end{equation}
to smooth maps:
\begin{equation}\label{mappini}
    \mathcal{S} \, : \, \mathbb{R}_t \, \rightarrow \, \mathcal{M}_g
\end{equation}
from the time real line $\mathbb{R}_t $ to a smooth Riemaniann
manifold $\mathcal{M}_g$ endowed with a metric $g$. The classical
case corresponds to $\mathcal{M} \, = \, \mathbb{R}^3$, $g_{ij}(x)
\, = \, \delta_{ij}$, but any other Riemaniann three-manifold might
be used and there exist generalization also to higher dimensions.
Adopting this point of view, the velocity field $\mathbf{u}\left( x
\, , \,t\right)$ is turned into a time evolving vector field on
$\mathcal{M}$ namely into a smooth family of sections of the tangent
bundle $\mathcal{T}\mathcal{M}$:
\begin{equation}\label{Ufildo}
    \forall \, t \, \in \, \mathbb{R} \, : \, u^i(x,t) \, \partial_i \,
    \equiv \, {\mathrm{U}}(t) \, \in \, \Gamma\left(\mathcal{T}\mathcal{M}\, , \, \mathcal{M}\right)
\end{equation}
Next, using the Riemaniann metric, which allows to raise and lower
tensor indices, to any ${\mathrm{U}}(t)$ we can associate a family
of sections of the cotangent bundle $\mathcal{T}^\star\mathcal{M}$
defined by the following time evolving one-form:
\begin{equation}\label{Omfildo}
  \forall \, t \, \in \, \mathbb{R} \, : \,    \Omega^{[\mathrm{U}]}(t)
  \, \equiv \, g_{ij} \, u^i(x,t) \, dx^j \, \in \, \Gamma\left(\mathcal{T}^\star\mathcal{M}\, , \, \mathcal{M}\right)
\end{equation}
Utilizing the exterior differential and the contraction operator
acting on differential forms, we can evaluate the Lie-derivative of
the one-form $\Omega^{[\mathrm{U}]}(t)$ along the vector field
$\mathrm{U}$. Applying definitions (see for instance
\cite{Fre:2013ika}, chapter five, page 120 of volume two) we obtain:
\begin{eqnarray}\label{trivialone}
    \mathcal{L}_\mathrm{U} \Omega^{[\mathrm{U}]}(t) & \equiv & {\rm i}_\mathrm{U}  \cdot {\rm d}
    \Omega^{[\mathrm{U}]} \, + \, {\rm d} \left({\rm i}_\mathrm{U} \cdot \Omega^{[\mathrm{U}]} \right) \nonumber\\
                                      & = &\left ( u^\ell \partial_\ell \, u^i  \, + \, g^{i k}
                                      \partial_{k} \, \underbrace{\parallel \mathrm{U} \parallel^2}_{g_{mn}
                                      \, u^m \, u^n} \right)\, g_{ij} \, dx^j
\end{eqnarray}
and the Navier Stokes  equation can be rewritten in either one of
the following two equivalent index-free reformulations:
\begin{eqnarray}
    - \, {\rm d } \left( p \, - \, \ft 12 \,\parallel \mathrm{U} \parallel^2\right ) & = &
    \partial_t \Omega^{[\mathrm{U}]} \, + \,  \mathcal{L}_\mathrm{U} \Omega^{[\mathrm{U}]} \, - \,\nu \,  \Delta_g \, \Omega^{[\mathrm{U}]}
    \, - \, \mathbf{f} \label{EulerBezIndex}\\
    &\mbox{or}& \nonumber\\
    - \, {\rm d } \left( p \, + \, \ft 12 \,\parallel \mathrm{U} \parallel^2\right ) & = &
    \partial_t \Omega^{[\mathrm{U}]} \, + \,  {\rm i}_\mathrm{U}
    \cdot {\rm d}  \Omega^{[\mathrm{U}]} \,  \, - \,\nu \,  \Delta_g \, \Omega^{[\mathrm{U}]}
    \, - \, \mathbf{f} \label{bernullini}
\end{eqnarray}
Where $\Delta_g$ is  the Laplace-Beltrami operator on 1-forms,
written in an index free notation as it follows:
\begin{equation}\label{laplacciobeltramo}
    \Delta_g \, = \, \delta\, \mathrm{d} \, + \, \mathrm{d} \, \delta \quad ;
    \quad\delta \equiv \star_g \, \mathrm{d} \star_g
\end{equation}
where with $\star_g$ we have denoted the Hodge duality operation in
the background of the metric $g$.
\par
Eq.(\ref{bernullini}) is one of the possible formulations of
classical Bernoulli theorem. To begin with, consider inviscid fluids
($\nu \, = \, 0$) with no external forces ($\mathbf{f} = 0$). Then
equation eq.(\ref{bernullini}) becomes:
 \begin{equation}\label{bernullone}
    - \, {\rm d } \left( p \, + \, \ft 12 \,\parallel \mathrm{U} \parallel^2\right ) \, =
    \,
    \partial_t \Omega^{[\mathrm{U}]} \, + \,  {\rm i}_\mathrm{U}
    \cdot {\rm d}  \Omega^{[\mathrm{U}]} \,
 \end{equation}
and from eq.(\ref{bernullini}) we immediately conclude that
\begin{equation}\label{finocchionabiscotta}
    H_B \, = \, p \, + \, \ft 12 \,\parallel \mathrm{U} \parallel^2
\end{equation}
is constant along the trajectories defined by eq.
(\ref{streamlines}). Turning matters around we can say that in
\textbf{steady flows} of inviscid free fluids, where
\begin{equation}\label{stedone}
    \partial_t \,\Omega^{[\mathrm{U}]} \, = \,0
\end{equation}
the fluid trajectories necessarily lay on the level surfaces
$H_B(\mathrm{x})\, = \, h \, \in \, \mathbb{R}$ of the function:
\begin{equation}\label{ignobilia}
    H \, : \, \mathcal{M} \, \rightarrow \, \mathbb{R}
\end{equation}
defined by (\ref{finocchionabiscotta}) and hereafter named, as it is
traditional in Fluid Mechanics, the \textbf{Bernoulli function}.
\par
An identical conclusion can be reached in the case of non vanishing
viscosity if the steady flow condition (\ref{stedone}) is replaced
by:
\begin{equation}\label{genstedona}
    \partial_t \,\Omega^{[\mathrm{U}]} \, = \,\nu \, \Delta_g\, \Omega^{[\mathrm{U}]} \, + \, \mathbf{f}
\end{equation}
For instance if at time $t=t_0$, the $1$-form
$\Omega^{[\mathrm{U}]}$ is the superposition of a collection of $N$
eigenstates of the Laplace-Beltrami operator:
\begin{equation}\label{zeroide}
    \Omega^{[\mathrm{U}]}\mid_{t=t_0}\, = \, \sum_{i=1}^N \, \omega_{i} \quad ;
    \quad \Delta_g\,\omega_{i}\, = \, \lambda_i \, \omega_i
\end{equation}
choosing a subset of such forms, say those from $i=1$ to $i=M < N$,
one can solve the condition (\ref{genstedona}) by setting the
driving force as follows:
\begin{equation}\label{laterano}
    \mathbf{f} \, = \, -\, \nu \, \sum_{i=1}^{M} \lambda_i \, \omega_i
\end{equation}
and the 1-form flow as follows:
\begin{equation}\label{carnitina}
\Omega^{[\mathrm{U}]}\, = \, \sum_{i=1}^{M} \, \omega_i \, +
\sum_{i=M+1}^N \, \omega_i \, \exp\left[-\lambda_i \, t\right]
\end{equation}
For viscid fluids, flows satisfying eq.(\ref{genstedona}) will be
referred to as \textbf{generalized steady flows}. It follows that in
the case of  steady and generalized steady flows the fluid
trajectories necessarily lay on the level surfaces
$H_B(\mathrm{x})\, = \, h \, \in \, \mathbb{R}$ of the Bernoulli
function (\ref{ignobilia}) defined by (\ref{finocchionabiscotta}).
\vskip 0.2cm
\subsection{Foliations} Then if $H_B(\mathbf{x})$ has a non trivial $x$-dependence
it defines a natural foliation of the $n$-dimensional manifold
$\mathcal{M}$ into a smooth family of $(n-1)$-manifolds (all
diffeomorphic among themselves) corresponding to the level surfaces.
Furthermore, as already advocated, the trajectories, \textit{i.e.}
the solutions of eq.(\ref{streamlines}), lay on these surfaces. In
other words the dynamical system encoded in eq.(\ref{streamlines})
is effectively $(n-1)$-dimensional admitting $H$ as an additional
conserved hamiltonian. In the classical  case $n\, = \, 3$ this
means that the differential system (\ref{streamlines}) is actually
two-dimensional, namely non chaotic and in some instances even
integrable\footnote{Here we rely on a general result established by
the theorem of Poincar\'e-Bendixson \cite{PoincareNonchaos,bendix}on
the limiting orbits of planar differential systems  whose corollary
is generally accepted to establish e that two-dimensional continuous
systems cannot be chaotic.}. Consequently we reach the conclusion
that no chaotic trajectories (or streamlines) can exist in those
domains where the Bernoulli function $H_B(x)$ has a non trivial
$x$-dependence: the only window open for lagrangian chaos occurs in
those domains where $H_B$ is a constant function. Looking at
eq.s(\ref{bernullini}-\ref{bernullone}) we realize that in steady
and generalized steady flows, the only open window for chaotic
trajectories is provided by velocity fields that satisfy the
condition:
\begin{equation}\label{cinesinotulipano}
    {\rm i}_\mathrm{U}  \cdot {\rm d}  \Omega^{[\mathrm{U}]} \, = \, 0
\end{equation}
This weak condition (\ref{cinesinotulipano}) is certainly satisfied
if the velocity field $\mathrm{U}$ satisfies the following strong
condition that is named \textbf{Beltrami equation}:
\begin{equation}
    {\rm d}  \Omega^{[\mathrm{U}]} \, = \, \lambda \, \star_g \, \Omega^{[\mathrm{U}]} \quad \Leftrightarrow \quad
   \star_g \, {\rm d}  \Omega^{[\mathrm{U}]} \, = \, \lambda \,  \Omega^{[\mathrm{U}]} \label{Belatramus}
\end{equation}
where $\star_g$, as already specified, denotes the Hodge duality
operator in the metric $g$:
\begin{eqnarray}\label{gongo}
    \star_g \, \Omega^{[\mathrm{U}]} &  = & \epsilon_{\ell m n} \, g^{\ell k}\, \Omega^{[\mathrm{U}]}_k \, dx^m \, \wedge \,dx^n
    \, =\,  u^\ell \, dx^m \, \wedge \,dx^n \, \epsilon_{\ell m n}\label{gruscia1}\\
    \star_g \, \mathrm{d} \, \Omega^{[\mathrm{U}]} &  = & \epsilon_{\ell m n} \, g^{m p}\,g^{n q}
    \partial_p \left( g_{qr} u^r\right)  \, dx^\ell \label{gruscia2}
\end{eqnarray}
\vskip 0.2cm
\subsection{Arnold theorem} The heuristic argument
which leads to  consider velocity fields that satisfy
\textit{Beltrami condition} (\ref{Belatramus}) as the unique steady
candidates compatible with chaotic trajectories was transformed by
Arnold into a rigorous theorem \cite{ArnoldBook} which, under the
strong hypothesis that $(\mathcal{M},g)$ is a closed, compact
Riemaniann three-manifold, states the following:
\begin{teorema}
\label{fundarnoltheo} (\textbf{Arnold}) There are only two
possibilities:
\begin{description}
  \item[a)] Either the form $\Omega^{[\mathrm{U}]}$ is an eigenstate of the Beltrami
  operator $\star_g \, \mathrm{d}$ with a non vanishing eigenvalue $\lambda \ne 0$
  \item[b)] or the manifold $\mathcal{M}$ is subdivided into a finite collection of cells,
  each of which admits a foliation diffeomorphic to $\mathrm{T^2} \times \mathbb{R}$ and every two-torus $T^2$
  is an invariant set with respect to the action of the velocity field $\mathrm{U}$: in other words,
  all trajectories lay on some $\mathrm{T^2}$  immersed in the manifold $\mathcal{M}$.
\end{description}
\end{teorema}
\textit{Chaotic trajectories or streamlines} are of particular
interest, both from the point of view of  theory and of
applications, since, in many scenarios, chaotic flows are  desirable
in order either to homogenize the heath exchange between the fluid
and some external objects immersed in the flow, like it happens in
\textit{autoclavs}, or to promote the mixing of two different
fluids, like it happens in \textit{chemical reactors}. The examples
are multiple and the mentioned ones are just an illustration.
\par
On the other hand the chaotic trajectories are desirable in all
these applications at \textit{small scales} while on \textit{larger
scales} the fluid should appear as moving steadily in some given
direction. The intrinsic non linearity of the NS equation forbids
the linear combination of solutions as new solutions and the
superposition of different regimes at different scales is a very
difficult mathematical problem that requires specialized analysis.
\par
The desire to investigate the on-set of chaotic trajectories in
steady (or generalized steady) flows of incompressible fluids
motivated the interest of the  dynamical system community in
Beltrami vector fields defined by  the condition (\ref{Belatramus}).
Furthermore, in view of the above powerful theorem proved by Arnold,
the focus of attention concentrated on the mathematically very
interesting case  of compact three-manifolds. Within this class, the
most easily treatable case is that of flat compact manifolds without
boundary, so that the most popular playground turned out to be the
three torus $\mathrm{T^3}$, whose possible role in applications has
already been emphasized.  Certainly many physical contexts for fluid
dynamics do not correspond to the idealized situation of a motion in
a compact manifold or, said differently, periodic boundary
conditions are not the most appropriate to be applied either in a
river, or in the atmosphere or in the charged plasmas environing a
compact star, yet the message conveyed by Arnold theorem that
Beltrami vector fields play a distinguished role in chaotic behavior
is to be taken seriously into account and gives an important hint.
 In view of what we are going to discus in  section
\label{viacontattosa} this hint is properly developed by considering
the one-to-one relation between Beltrami fields and contact
structures on three-manifolds that is now extended to contact
structures with singularities. Before coming to that we dwell on a
short review of \textit{a few} of \textbf{the very few known exact
solutions} of Navier Stokes equations whose landscape is on one hand
quite variegated while on the other hand shows the difficulty of
finding solutions of chaotic type.
\section{Review of a few exact
solutions of the Navier Stokes equations} \label{passerella} Some of
the known solutions are quite old, some are newer, some are
unsteady, some are steady. They were all found by means of some
educated mathematical ansatz coupled with intuition about certain
physical-mechanical scenarios.
\subsection{The gradient or potential
flow}\label{potenzialflusso} A very general class of steady
solutions is provided by the so named potential flows. Let
$\phi(\mathbf{x})$ be some smooth function of the space coordinates.
If we set the velocity field $\mathrm{U}$ equal to the gradient of
$\phi$:
\begin{equation}\label{gradientflow}
    U^i \, = \, g^{ij}\, \partial_j \phi
\end{equation}
we immediately obtain that the one-form $\Omega^{[U]} \, = \,
\mathrm{d}\phi$ is the exterior differential of the potential
$\phi$. This immediately implies that $\mathrm{d}\Omega^{[U]}\,=\,
\mathrm{d}^2 \phi \, = \, 0$ and the critical non linear term
$i_U\Omega^{[U]}$ is canceled from equation (\ref{bernullini}).
Assuming the Bernoulli function constant, namely setting the
pressure equal to:
\begin{equation}\label{pressione}
    p= -\frac{1}{2} \parallel\nabla\phi\parallel^2 +k
\end{equation}
eq. (\ref{bernullini}) reduces to:
\begin{equation}\label{curiosone}
    \,\nu \, \Delta_g \, \mathrm{d}\phi \, = \,0
\end{equation}
For inviscid fluids ($\nu \, = \, 0$) the equation is satisfied with
any generic potential $\phi$. If the viscosity does not vanish we
need to satisfy the equation:
\begin{equation}\label{lopiatto}
    \Delta_g \,\mathrm{d}\phi \, = \, 0
\end{equation}
The formal structure (\ref{laplacciobeltramo}) of the
Laplace-Beltrami operator turns out very useful in this respect
because it immediately shows that:
\begin{equation}\label{lopiatto}
    \Delta_g \,\mathrm{d}\phi \, = \,\mathrm{ d}\Delta_g \,\phi
\end{equation}
Hence in order to satisfy the equation also in the case $\nu \neq 0$
it is sufficient that:
\begin{equation}\label{pescione}
    \Delta_g \,\phi \, = \, \mathfrak{c} \, = \, \text{const}
\end{equation}
Eq. (\ref{pescione}) is just Poisson equation that has a large
variety of solutions depending on the assume symmetry. We just give
an example of axis symmetric solution. Let us assume that the
potential $\phi$ depends only on $r=\sqrt{x^2 + y^2}$ and the third
coordinate $z$. A general solution for the Poisson problem in this
case is given by:
\begin{equation}\label{canocchio}
    \phi(x,y,z) \, = \, \phi\left(r,z\right)\, = \,\frac{\kappa  r^2}{4}+\log (r) (\delta +\gamma  z)+\frac{\alpha  z^2}{2}+\beta  z
\end{equation}
Where $\alpha,\beta,\gamma,\delta,\kappa$ are constant parameters
that can be arbitrarily chosen and the constant mentioned in eq.
(\ref{pescione}) is $\mathfrak{c}\,=\, \kappa+\alpha$. Playing with
these parameters one can obtain very different flows but always
\textit{irrotational} and far from being chaotic. Just for
illustration we choose the following case:
\begin{equation}\label{casopotuno}
    \phi_{0}(x,y,z)\, = \,-\frac{r^2}{4}+\frac{1}{5} z \log (r)-\frac{z^2}{4}+\frac{z}{2}
\end{equation}
and we obtain the vector field plotted in fig.\ref{ballavantana}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=90mm]{vettofonte.png}
\end{center}

\caption{\it  Plot of the axis symmetric gradient vector field
$\nabla\phi$ where $\phi$ is given in eq.(\ref{casopotuno}). As we
see, far from the central axis the vector field is always attractive
versus the center. Yet coming close to the center the logarithmic
part of the potential diverges and gives repulsion for $z>0$ while
it reinforce the attraction for $z<0$.} \label{ballavantana}

 \hskip 1cm \unitlength=1.1mm

\end{figure}
In fig.\ref{fontanina} we show a family of streamlines of this flow.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=55mm]{fontanadasopra.png}
\includegraphics[height=55mm]{fontanadisbieco.png}
\includegraphics[height=55mm]{fontanadifronte.png}
\end{center}

\caption{\it  A family of streamlines for the potential flow with
$\phi$ given by eq.(\ref{casopotuno}) is shown from above (left),
from the a side (center) and in frontal view (right)}
\label{fontanina}

 \hskip 1cm \unitlength=1.1mm

\end{figure}
Finally in fig.\ref{conogelato} we show the foliation of three
dimensional space induced by the level surfaces of the potential
$\phi_0(x,y,y)=h$. In fig.\ref{fontanina} we show a family of
streamlines of this flow.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=120mm]{gelatoallamela.png}
\end{center}

\caption{\it The foliation induced by the level surfaces of the
potential $\phi_0$ given by eq.(\ref{casopotuno}). In each point of
3D space the flow is orthogonal to the tangent space to the level
surface passing through that point.} \label{conogelato}

 \hskip 1cm \unitlength=1.1mm

\end{figure}
\subsection{The Poiseuille flow}
\label{pescheria} Next we present a steady solution of the Navier
Stokes equations, named after Poiseuille that well describes the
flow of a fluid in a cylindrical tube, induced by a difference of
pressure at the beginning  and at the end of the tube. Once again
this solution relies on a solution of the 2D Poisson equation.
\par
We make the following ansatz for the velocity vector and for the
pressure:
\begin{eqnarray}
\label{pescetto}
  \mathbf{U} &=& \left\{0,0,f(x,y)\right\}\nonumber\\
  p &=& p(z) = a \, z \,+\, b
\end{eqnarray}
where $a,b$ are constants to be determined and $f(x,y)$ is a so far
undetermined function of the coordinates in the plane orthogonal to
direction of the fluid velocity. Indeed this latter moves steadily
in the $z$-direction with a velocity that depends only on the
position on that orthogonal plane. Clearly this ansatz models the
physical situation of a fluid flowing in a tube under the drive of a
pressure that is uniform on the cross-section of the tube, but
different at different distances from some initial point. The main
point about the ansatz (\ref{pescetto}) is that neutralizes the non
linear term setting it to zero like in the case of the potential
flow:
\begin{equation}\label{vianonlin}
    \mathbf{U}\cdot\nabla \mathbf{U}\, \equiv
    \,i_{\mathbf{U}}\mathrm{d}\Omega^{[\mathbf{U}]} \, = \, 0
\end{equation}
Assuming the flow to be steady the entire NS equation
(\ref{EulerusEqua}) reduces, in the absence of external forces to:
\begin{equation}\label{connubio}
    \nu \, \Delta_2 f(x,y)\, - \,\frac{\partial_z p(z)}{\partial z} \,
    = 0 \quad \Rightarrow \quad \Delta_2 f(x,y) \, = \,
    \frac{a}{\nu} \, = \, \text{const}
\end{equation}
where $\Delta_2 \, = \, \frac{\partial^2}{\partial x^2}
+\frac{\partial^2}{\partial y^2}$ is the two-dimensional laplacian
operator and the second equation in (\ref{connubio}) is Poisson
equation. We can easily find $\mathrm{SO(2)}$ invariant solutions of
such an equation setting:
\begin{equation}\label{soluso2}
    f(x,y) \, = \, \frac{R^2 -x^2-y^2}{\sqrt{2}}
\end{equation}
where $R$ is a constant that can be interpreted as the radius of a
circular tube hosting the flow and the denominator $\sqrt{2}$ is
just a convenient normalization when we shall compare the classical
Poiseuille solution with Beltrami flows obtained from the hexagonal
lattice compactification of $\mathbb{R}^3$. We fulfill eq.
(\ref{connubio}) by setting $a=-2\sqrt{2}\nu$, which tells us that
the larger is the viscosity the faster is the linear decay of  the
pressure along the tube.  Indeed we can interpret the constant $b$
in the second of eq.s(\ref{pescetto}) as the value of the pressure
at the beginning of the tube. At the distance
$z_0=\frac{b}{2\sqrt{2}\nu}$ the pressure vanishes and further on it
becomes negative. Hence at the $z=z_0$ the flow stops and the length
of the flow $[0,z_0]$ shrinks to shorter and shorter when the
viscosity is increased. All the streamlines are parallel and
uniformly distributed around the central axis of the tube, with
rotational symmetry. The length of the streamlines is maximum at the
center of the tube and decreases to zero as you move towards the
enveloping wall of the cylindrical tube as it is shown in fig.
\ref{caledonia}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=90mm]{tubonepesce.png}
\end{center}

\caption{\it Streamlines of the Poiseuille flow in a cylindrical
tube with axial rotational symmetry.} \label{caledonia}

 \hskip 1cm \unitlength=1.1mm

\end{figure}
\subsubsection{Hexagonal Poiseuille Flow} The Poisson equation
(\ref{connubio}) does not need to be necessarily solve with axial
rotational symmetry. There are other solutions. One interesting
choice is that of requiring only a discrete rotational symmetry. For
future use in connection with Beltrami flows on
$\mathrm{T^3}/\Lambda_{hex}$ where $\Lambda_{hex}$ is the hexagonal
lattice, we consider the solution of Poisson equation requiring only
invariance with respect to rotations of an angle $\theta=2\pi/6$
around the central axis. Referring to the basis vectors
$\mathbf{w}_{1,2,3}$ of the hexagonal lattice, presented later in
eq.(\ref{Hexagbase}) of section \ref{hexareticolo}, we trade the
coordinates $\{x,y,z\}$ for the coordinates $u,v,w$ defined by the
equation:
\begin{equation}\label{nuovabase}
    \{x,y,z\} \, = \, u \,\mathbf{w}_1 \, + v\,\mathbf{w}_2+
    w\,\mathbf{w}_3
\end{equation}
The two coordinates of the $XY$ plane are functions only of $u,v$
and we have:
\begin{equation}\label{radicchiogiallo}
  x\, = \, x(u,v) \, \equiv \,  \frac{2 u-v}{\sqrt{2}}, \,\,\, y\, y(u,v) \, \equiv \, \sqrt{\frac{3}{2}}
  v \quad \Leftrightarrow \quad u\,
  =\,\frac{x}{\sqrt{2}}+\frac{y}{\sqrt{6}},\,\,\, v \, = \, \sqrt{\frac{2}{3}} y
\end{equation}
In terms of the $u,v$ coordinates the Laplace operator $\Delta_2$
acting on $f(x,y)\equiv \Phi(u,v)$ takes the form:
\begin{equation}\label{uvlaplac}
    \Delta_2 \Phi(u,v) \, = \,\frac{2}{3} \left(\partial_u^2\,\Phi(u,v)+\partial_u\partial_v \Phi(u,v)+\partial^2_v\Phi(u,v)\right)
\end{equation}
Instead of imposing the condition of circular symmetry as in the
previous discussion we impose on $\Phi(u,v)$ the condition of being
invariant with respect to a discrete $\mathbb{Z}_6$ group provided
by rotations of an angle $\pi/3$ and its multiples. In the $u,v$
basis such a rotation is represented by the following integer valued
matrix:
\begin{equation}\label{Arotone}
 \mathcal{ A}  \, = \, \left(
\begin{array}{cc}
 1 & 1 \\
 -1 & 0 \\
\end{array}
\right)
\end{equation}
Hence the invariance condition to be imposed on $\Phi(u,v)$ is the
following one:
\begin{equation}\label{invariantu6}
    \Phi(u+v,-u) \, = \, \Phi(u,v)
\end{equation}
Assuming $\Phi(u,v)$ to be a second order polynomial, as it is
necessary in order to satisfy Poisson equation, equation
(\ref{invariantu6}) has a unique solution:
\begin{equation}\label{bernacco}
 \Phi(u,v) \, = \, \Psi_{(\alpha,\beta)}(u,v) \, \equiv \,   \alpha  \left(-\beta +u^2+u v+v^2\right)
\end{equation}
where $\alpha,\beta$ are constant parameters. The velocity field in
the standard basis is therefore given by:
\begin{equation}\label{velocitone}
    \mathbf{U}\, = \, \left\{ 0,0,\Psi_{(\alpha,\beta)}(u,v)\right\}
\end{equation}
and every streamline is a straight line starting  at
$\{x(u,v),y(u,v),0\}$ at time $t=0$ and ending in
$\{x(u,v),y(u,v),\Psi_{(\alpha,\beta)}(u,v)\times \tau\}$ at time
$t=\tau$. We consider the surface described by such endpoints for
the fundamental cell given by $0\leq u \leq 1,\,\,0\leq v \leq 1$
and we obtain the picture displayed on the left of
fig.\ref{portamelamarcia}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=90mm]{apereginacella.png}
\includegraphics[width=90mm]{fucocompleto.png}
\end{center}

\caption{\it Streamlines of the Poiseuille flow in a hexagonal tube
 with $\mathbb{Z}_6$ symmetry. On the left we see the surface traced by the end points of the streamlines starting in
 one of the three fundamental lattice cells of which the hexagon is composed. This latter is marked in red at the bottom.
 rotating all the stream-lines by an angle $2\pi/3$ we obtain the other two sectors of the fluid profile (picture on the right)}
 \label{portamelamarcia}

 \hskip 1cm \unitlength=1.1mm

\end{figure}
Then it suffices to rotate all the streamlines by an angle $2\pi/3$
and  once more by the same angle and we obtain the complete profile
of the flow displayed in the picture on the right of
fig.\ref{portamelamarcia}.
\subsection{Generalized Beltrami flows}
\label{genbeltraflow} Another class of steady flow solutions of the
Navier Stokes equations, that may lead to vortices but never to
chaotic behavior are the so named \textbf{generalized Beltrami
flows} that in classical fluid dynamics literature (see for example
\cite{wangunsteady}) are those characterized by the condition:
\begin{equation}\label{carnadino}
    \nabla \times \left[\left( \nabla \times \mathbf{U}\right ) \times
    \mathbf{U}\right] \, = \, 0
\end{equation}
where according to old-fashioned three-dimensional tensor calculus
$\times$ denotes the vector-product and $\nabla \times $ the
curl-operation. In differential form language equation
(\ref{carnadino}) can be rewritten as:
\begin{equation}\label{carnadino}
    \star_g \mathrm{d} \star_g \left ( \star_g \mathrm{d}\Omega^{[U]} \wedge \Omega^{[U]}\right) \, = \,
    \delta \left ( \star_g \mathrm{d}\Omega^{[U]} \wedge
    \Omega^{[U]}\right) \, = \, 0
\end{equation}
namely the $2$-form:
\begin{equation}\label{tettaforma}
    \Theta^{[U]} \, \equiv \,\left ( \star_g
\mathrm{d}\Omega^{[U]} \wedge \Omega^{[U]}\right)
\end{equation}
is required to be co-closed, namely to be in the kernel of the
adjoint  exterior differential which also squares to zero:
$\delta^2=0$. The reason why such flows are named
\textit{generalized Beltrami} is that for Beltrami flows the 2-form
is not only co-closed, it just vanishes since $\star_g
\mathrm{d}\Omega^{[U]} \, \propto \, \Omega^{[U]}$. Among the
generalized Beltrami solutions a subclass is provided by
\textit{planar models}.
\subsubsection{Planar Models} To make a long
story short the planar generalized Beltrami solutions are those
where the ansatz for the velocity field is the following one:
\begin{equation}\label{carduccio}
    \mathbf{U}=\left\{\partial_y \Psi(x,y), \, -\partial_x
    \Psi(x,y),\, 0\right\}
\end{equation}
the function $\Psi(x,y)$ of the two planar coordinates $x,y$ being
given the name, in Fluid Dynamics literature, of \textbf{stream
function}. With such an ansatz the NS equations (\ref{EulerusEqua})
reduce to:
\begin{eqnarray}\label{crocediferro}
  - \, \partial_y^2\Psi \, \partial_x\Psi + \partial_x\partial_y\Psi \, \partial_y \Psi \, - \,
  \nu \left( \partial_y^3\Psi +\partial^2_x\partial_y \Psi\right) +\partial_x \, p &=& 0 \nonumber\\
- \, \partial_x^2\Psi \, \partial_y\Psi + \partial_x\partial_y\Psi
\, \partial_x \Psi \, + \,
  \nu \left( \partial_x^3\Psi +\partial^2_y\partial_x \Psi\right) +\partial_y \, p &=& 0
  \nonumber\\
  \partial_z \, p &=& 0
\end{eqnarray}
Equations (\ref{crocediferro}) that immediately imply
$z$-independence also of the pressure $p$  have been solved in
various ways at different time with interesting ans\"{a}tze.
\subsubsection{Vortex in Shear Flow} An example of solution, within
the scope of the planar ansatz, is obtained by choosing the
following stream function:
\begin{equation}\label{shearvort}
   \Psi \, = \,  \Psi_{SV}(x,y) \, = \,\alpha  y\, + \,\beta  y^2 \, + \,\gamma  \arctan\left(\frac{y}{x}\right)
\end{equation}
where $\alpha,\beta,\gamma$ are real constants. Inserting the
function $\Psi_{SV}(x,y)$ into eq.s (\ref{crocediferro}) we get what
follows:
\begin{eqnarray}
 \partial_x \mathfrak{p} \, +\,\frac{\gamma  \left(-\alpha  x^2-\gamma  x+y^2 (\alpha +4
   \beta  y)\right)}{\left(x^2+y^2\right)^2}&=& 0 \nonumber \\
\partial_y \mathfrak{p} \, -\,\frac{\gamma  y (\gamma +2 x (\alpha +2 \beta
   y))}{\left(x^2+y^2\right)^2}&=& 0\label{bocciodicannella}
\end{eqnarray}
As we see the viscosity $\nu$ has disappeared from the equations.
This means that the flow we are constructing is a so named
\textbf{universal flow} possible for any value of the viscosity.
What remains in order to obtain integration of the partial
differential equations is to find a single function, if it exists,
$\mathfrak{p}(x,y)$ whose partial derivatives are those specified by
eq.s (\ref{bocciodicannella}). The answer is yes and the pressure is
given by:
\begin{equation}\label{lotodoro}
    \mathfrak{p}(x,y)\, = \, \gamma  \left(\frac{\gamma +2 x (\alpha +2 \beta  y)}{2 \left(x^2+y^2\right)}-2 \beta
   \arctan\left(\frac{y}{x}\right)\right) \, + \, \text{const}
\end{equation}
Explicitly the velocity field is the following one:
\begin{equation}\label{caluno}
  \mathbf{U} \, =\,  \left\{\alpha +\frac{\gamma  x}{x^2+y^2}+2 \beta  y,\frac{\gamma
   y}{x^2+y^2},0\right\}
\end{equation}
Its plot is displayed in fig.\ref{alieno}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=60mm]{shearflow.png}
\includegraphics[width=60mm]{alettavetplot.png}
\end{center}

\caption{\it On the left we display streamlines of the shear flow
whose velocity field is in eq. (\ref{caluno}). The choice of
parameters is $\alpha=4$,$\beta=-1/3$,$\gamma=-1/2$.  On the right
we see a plot the velocity vector field.}
 \label{alieno}

 \hskip 1cm \unitlength=1.1mm

\end{figure}
\subsubsection{The Elliptic Vortex} The last example that we present
of generalized Beltrami planar flow is obtained by choosing the
following stream function:
\begin{equation}\label{ellisso}
   \Psi (x,y)= \alpha  x^2+\beta  y^2
\end{equation}
In this case the NS equations \ref{crocediferro} are satisfied if we
use the following function for the pressure:
\begin{equation}\label{ganimede}
   \mathfrak{ p}(x,y)\, =\, 2 \alpha  \beta  \left(x^2+y^2\right)
\end{equation}
and the velocity field is:
\begin{equation}\label{scappascappa}
    \mathbf{U}\, = \, \left\{2 \beta  y,-2 \alpha  x,0 \right\}
\end{equation}
In this case the differential system for the streamlines is exactly
integrable and we obtain:
\begin{eqnarray}
\label{pernacchione}
  x(t) &=& a \cos \left(2 \sqrt{\alpha } \sqrt{\beta }\, t\right)+b \sin \left(2 \sqrt{\alpha }
   \sqrt{\beta }\, t\right) \nonumber \\
  y(t) &=& \frac{\sqrt{\alpha } \left(b \cos \left(2 \sqrt{\alpha } \sqrt{\beta } \,t\right)-a \sin
   \left(2 \sqrt{\alpha } \sqrt{\beta }\, t\right)\right)}{\sqrt{\beta }}
\end{eqnarray}
Stremlines and plot of the vector field \ref{scappascappa}) are
shown in fig.(\ref{vorticosamente}).
\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=60mm]{vortstream.png}
\includegraphics[width=60mm]{vortfield.png}
\end{center}

\caption{\it On the left we display streamlines of the planar
elliptic vortex flow whose velocity field is in eq.
(\ref{scappascappa}). On the right we display a vector plot of the
velocity field. The choice of parameters is $\alpha=3$,$\beta=5$.}
 \label{vorticosamente}

 \hskip 1cm \unitlength=1.1mm

\end{figure}
\section{The path leading to contact geometry}
\label{viacontattosa} Last but not least let us mention that
Beltrami vector fields are intimately related with the mathematical
conception of \textit{contact geometry and contact topology}.   As
we have seen from our sketch of Arnold Theorem, the main obstacle to
the onset of chaotic trajectories has a distinctive geometrical
flavor: trajectories are necessarily ordered and non chaotic if the
manifold where they take place has a foliated structure $\Sigma_h
\times \mathbb{R}_h$, the two dimensional level sets $\Sigma_h$
being invariant under the action of the velocity vector field $U$.
In this case each streamline lays on some surface $\Sigma_h$.
Equally adverse to chaotic trajectories is the case of
\textit{gradient flows} where there is  a foliation provided by the
level sets of some function $H(x)$ and the velocity field
$\mathrm{U}=\nabla H$ is just the gradient of $H$. In this case all
trajectories are orthogonal to the leaves $\Sigma_h$ of the
foliation and their well aligned tangent vectors are parallel to its
normal vector.
\par
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{disegnus2.jpg}
\end{center}

\caption{\it  Schematic view of the foliation of a three dimensional
manifold $\mathcal{M}$. The family of two-dimensional surfaces
$\Sigma_h$ are typically the level sets $H(\mathbf{x}) =h$ of some
function $H\, : \, \mathcal{M} \, \rightarrow\, \mathbb{R}$. At each
point of $p\, \in \, \Sigma_h \subset \mathcal{M}$ the dashed
vectors span the tangent space $T_p^\parallel\Sigma_h$, while the
solid vector span the normal space to the surface
$T_p^\perp\Sigma_h$. Equally adverse to chaotic trajectories is the
case where the velocity field $\mathrm{U}$  lies in
$T_p^\perp\Sigma_h$ (gradient flow) or in $T_p^\parallel\Sigma_h$ }
\label{fogliattone}

 \hskip 1cm \unitlength=1.1mm

\end{figure}
In conclusion in presence of a foliation we have the following
decomposition of the tangent space to the manifold $\mathcal{M}$ at
any point $p \in \mathcal{M}$
 \begin{equation}\label{finchius}
    T_p\mathcal{M} \, = \, T_p^\perp\Sigma_h \, \oplus \, T_p^\parallel\Sigma_h
\end{equation}
and no chaotic trajectories are possible in a region $\mathfrak{S}
\subset \mathcal{M}$ where $\mathrm{U}(p) \, \in \,
T_p^\perp\Sigma_h $ or $\mathrm{U}(p) \, \in \,
T_p^\parallel\Sigma_h$ for $\forall p \, \in \, \mathfrak{S} $ (see
fig.\ref{fogliattone}).
\par
This matter of fact motivates an attempt to capture the geometry  of
the bundle of subspaces orthogonal to the lines of flow by
introducing  an intrinsic topological indicator that distinguishes
necessarily non chaotic flows from possibly chaotic ones. Let us
first consider the extreme case of a gradient flow where
$\Omega^{[\mathrm{U}]} \, = \, \mathrm{d} H$ is an exact form. For
such flows we have:
\begin{equation}\label{canedellascala}
\Omega^{[\mathrm{U}]} \, \wedge \, \mathrm{d}\Omega^{[\mathrm{U}]}
\, = \, \Omega^{[\mathrm{U}]} \, \wedge \, \underbrace{\mathrm{d} \,
\mathrm{d}H}_{= \, 0} \, = \, 0
\end{equation}
Secondly let us consider the opposite case where the velocity field
$\mathrm{U}$ is orthogonal to a gradient vector field $\nabla H$ so
that the integral curves of $\mathrm{U}$  lay on the level surfaces
$\Sigma_h$. Furthermore let us assume that $\mathrm{U}$ is self
similar on neighboring level surfaces. We can characterize this
situation in a Riemaniann manifold $(\mathcal{M},g)$ by the
following two conditions:
\begin{equation}\label{gattoascensore}
{\rm i}_{\nabla H}\Omega^{[\mathrm{U}]} \, \Leftrightarrow \,
g\left(\mathrm{U}\, ,\, \nabla H \right) \, = \, 0 \quad ; \quad
\left[ U\, , \, \nabla H\right] \, = \, 0
\end{equation}
The first of eq.s(\ref{gattoascensore}) is obvious. To grasp the
second it is sufficient to introduce, in the neighborhood of any
point $p\, \in \, \mathcal{M}$, a local coordinate system composed
by $(h,x^\parallel )$ where $h$ is the value of the function $H$ and
$x^\parallel $ denotes some local coordinate system on the level set
$\Sigma_h$.  The situation we have described corresponds to assuming
that:
\begin{equation}\label{caruccio}
    \mathrm{U} \, \simeq \, U^{\parallel} (x^\parallel) \, \partial_\parallel  \quad ; \quad \partial_h
    \, U^{\parallel} (x^\parallel) \, = \, 0
\end{equation}
Under the conditions spelled out in eq.(\ref{gattoascensore}) we can
easily prove that:
\begin{equation}\label{suschione}
    {\rm i}_{\nabla H} \, \mathrm{d}\Omega^{[\mathrm{U}]} \, = \,0
\end{equation}
Indeed from the definition of the Lie derivative we obtain:
\begin{eqnarray}\label{cincischiando}
    {\rm i}_{\nabla H} \, \mathrm{d}\Omega^{[\mathrm{U}]} \, = \, \underbrace{\mathcal{L}_{\nabla H}
    \,\Omega^{[\mathrm{U}]}}_{= \, \Omega^{\left[[U\, ,\, \nabla H]\right]} \, = \, 0} \, - \,
    \mathrm{d}\left(\underbrace{{\rm i}_{\nabla H}\Omega^{[\mathrm{U}]}}_{= 0}\right)
\end{eqnarray}
Since we have both ${\rm i}_{\nabla H}\Omega^{[\mathrm{U}]} \, = \,
0$ and ${\rm i}_{\nabla H}\Omega^{[\mathrm{U}]} \, = \, 0$ it
follows that also in this case:
\begin{equation}\label{franceschiello}
    \Omega^{[\mathrm{U}]} \, \wedge \, \mathrm{d}\Omega^{[\mathrm{U}]} \, = \, 0
\end{equation}
Therefore in order not to exclude chaotic trajectories one has to
assume that
\begin{equation}\label{rebomba}
    \Omega^{[\mathrm{U}]} \, \wedge \, \mathrm{d}\Omega^{[\mathrm{U}]} \, \neq \, 0
\end{equation}
and the above condition is what leads us to \textit{contact
geometry}.
\chapter{Geometrical Foundations}
\section{Contact Geometry}
Contact Geometry is both an old and a relatively new chapter of
Mathematics, since it springs from some classical results of
analysis that date back to Darboux, Goursat, Lie and other XIX
century maitres, yet it has been vigorously developed in the last
two decades from a relatively small community of mathematicians. To
say it in short, \textit{Contact Geometry} is a mathematical theory
aiming at providing an intrinsic geometrical-topological
characterization of \textit{non integrability} and, in some sense,
it is framed within a perspective which is just the reversed one of
that customary in standard cohomological theories and in the theory
of integrable systems. Not surprisingly Contact Geometry comes into
play when the aim is that of \textit{chaos and disorder}, rather
than that of \textit{order}, like it happens in the framework of
\textit{integrable systems} where, by means of clever
transformations and procedures, \textit{non-linearity} is tamed and
the evolution of whatever is the object of study is reduced to
simplicity.
\par
Contact Geometry deals exclusively with real \textit{Differential
Manifolds $\mathcal{M}_{2n+1}$ of odd-dimension} and on the other
hand it has a symbiotic relation with  \textit{Symplectic Manifolds
$\mathcal{S}_{2n+2}$ and $\mathcal{S}_{2n}$} in the two adjacent
even dimensions, upper and lower.
\par
The fundamental notions of \textit{Contact Geometry} are easily
spelled out and, in order to be grasped, they do not require more
than the most elementary basic concepts of \textit{Differential
Geometry}. In the concise overview that we provide in this section
we closely follow the excellent review \cite{contatoregeiges}.
\subsection{Contact structures}
We consider an odd dimensional differential manifold
$\mathcal{M}_{2n+1}$ and its tangent bundle:
\begin{equation}\label{tangentobundolo}
    \mathcal{TM}_{2n+1} \, \stackrel{\pi}{\longrightarrow} \,
    \mathcal{M}_{2n+1} \quad ; \quad \forall \, p \,
    \in \, \mathcal{M}_{2n+1} \quad , \quad \pi^{-1}(p) \, \sim \,
    \mathbb{R}^{2n+1}
\end{equation}
By definition, the transition function between any two local
trivializations of the tangent bundle is just provided by the
\textit{inverse jacobian} of the transition  function $\psi_{ij}(x)$
between the two corresponding open charts $(U_i,\varphi_i)$ and
$(U_j,\varphi_j)$ in any atlas that covers the base manifold
$\mathcal{M}_{2n+1}$ (see for instance \cite{Fre:2013ika}).
\par
The space of sections of the tangent bundle
$\Gamma\left[\mathcal{TM}_{2n+1},\mathcal{M}_{2n+1}\right]$ is
composed by vector fields, whose local description is in terms of
first order differential operators:
\begin{eqnarray}\label{vettorifildi}
    &&\mathbf{X} \, \in \, \Gamma\left[\mathcal{TM}_{2n+1}\right] \quad
    \Downarrow \nonumber\\
    &&  \mathbf{X} \, = \, X^\mu \left(x\right)
    \,\frac{\partial}{\partial x^\mu} \quad : \quad \text{in any open patch $U$
    with coordinates $x^1\dots x^{2n+1}$}
\end{eqnarray}
The cotangent bundle
\begin{equation}\label{cotangentobundolo}
    \mathcal{T}^\star\mathcal{M}_{2n+1} \, \stackrel{\pi_\star}{\longrightarrow} \,
    \mathcal{M}_{2n+1} \quad ; \quad \forall \, p \,
    \in \, \mathcal{M}_{2n+1} \quad , \quad \pi_\star^{-1}(p) \, \sim \,
    \left(\mathbb{R}^{2n+1}\right)^\star
\end{equation}
is the dual of the tangent one, in the sense that its fibre over
each point $p$ of the base manifold is the dual vector space of the
fibre over the same point in the tangent bundle, \textit{i.e.} the
space of linear functionals over the latter:
\begin{equation}\label{cricco}
    \forall p \in \, \mathcal{M}_{2n+1} \quad : \quad
    \pi_\star^{-1}(p) \, = \, \text{space of linear functionals on}
    \, \pi^{-1}(p)
    \,
\end{equation}
By construction, the transition function between any two local
trivializations of the cotangent bundle is just provided by the
\textit{direct jacobian} of the transition function between the two
corresponding open charts $(U_i,\varphi_i)$ and $(U_j,\varphi_j)$,
in any atlas that covers the base manifold $\mathcal{M}_{2n+1}$.
\par
The space of sections of the cotangent bundle
$\Gamma\left[\mathcal{T}^\star\mathcal{M}_{2n+1},\mathcal{M}_{2n+1}\right]$
is composed by differential 1-forms, whose local description is
recalled below:
\begin{eqnarray}\label{vettorifildi}
    &&\omega \, \in \, \Gamma\left[\mathcal{T}^\star\mathcal{M}_{2n+1},\mathcal{M}_{2n+1}\right] \quad
    \Downarrow \nonumber\\
    &&  \omega \, = \, \omega_\mu \left(x\right)
    \,dx^\mu \quad \quad : \quad \text{In any open patch $U$
    with coordinates $x^1\dots x^{2n+1}$}
\end{eqnarray}
In terms of these fundamental concepts, that apply to manifolds of
any dimension, both even and odd, one can introduce the concept of
hyperplane bundles.
\par
A hyperplane bundle is a reduction of the tangent bundle where the
fibres over each point constitute a codimension one vector subspace
of the tangent space in the same point,  the transition functions
being accordingly derived:
\begin{eqnarray}\label{iperpiano}
    \mathcal{HY}\,
    \stackrel{\mathcal{P}}{\longrightarrow} \, \mathcal{M} & ;
    & \forall p \in \mathcal{M} \, , \,
    \mathcal{P}^{-1}(p)
    \subset\pi^{-1}(p) \quad \text{where} \quad \mathcal{T}\mathcal{M} \, \stackrel{\pi}{\longrightarrow} \,
    \mathcal{M}\nonumber\\
    \mathrm{dim}_\mathbb{R}\mathcal{M}\, = \, m & ;&\mathrm{dim}_\mathbb{R}\pi^{-1}(p) \, = \, m \quad ; \quad
    \mathrm{dim}_\mathbb{R}\mathcal{P}^{-1}(p) \, = \, m-1
\end{eqnarray}
A simple  way of constructing a hyperplane bundle is by means of the
choice of a section of the cotangent bundle namely of some 1-form
$\omega \in \Gamma\left[\mathcal{T}^\star
\mathcal{M},\mathcal{M}\right]$. Then the hyperplane sub-bundle
$\mathcal{HY}^\omega \subset \mathcal{TM}$ of the tangent bundle is
implicitly defined by stating what is the space of its sections
$\Gamma\left[\mathcal{HY}^\omega,\mathcal{M}\right]$, namely
mentioning all the possible vector fields that are sections of
$\mathcal{HY}^\omega$. Utilizing a precise mathematical language let
$\mathbf{X}\in \Gamma\left[\mathcal{TM},\mathcal{M}\right]$ be a
vector field, we write
\begin{equation}\label{kernello}
    \mathbf{X} \, \in \, \Gamma\left[\mathcal{HY}^\omega,\mathcal{M}\right] \quad
    \text{iff} \quad \mathbf{X} \, \in \,\mathrm{ker}\, \omega \quad
    \textit{i.e.} \quad \omega\left(\mathbf{X}\right) \equiv 0 \quad
    (\text{everywhere})
\end{equation}
\begin{definizione}
Given a manifold $\mathcal{M}_{2n+1}$ of odd dimension, a
\textbf{contact structure} on $\mathcal{M}_{2n+1}$ is a rank $2n$
sub-bundle $\xi \, \stackrel{\mathcal{P}}{\longrightarrow}\,
\mathcal{M}_{2n+1}$ of the tangent bundle $\mathcal{TM}_{2n+1} \,
\stackrel{\pi}{\longrightarrow}\, \mathcal{M}_{2n+1}$ that can be
identified with the hyperplane bundle $\mathcal{HY}^\alpha$ where
the 1-form $\alpha$ satisfies the following condition:
\begin{equation}\label{latisanaditiglio}
    \alpha \, \wedge \, \underbrace{\mathrm{d}\alpha \, \wedge \,\dots \, \, \wedge
    \,\mathrm{d}\alpha}_{n-\text{times}} \, \neq \, 0 \quad
    (\text{everywhere on $\mathcal{M}_{2n+1}$})
\end{equation}
The $1$-form $\alpha$ is named a \textbf{contact form}.
\end{definizione}
\begin{definizione}
A \textbf{contact manifold} is a pair $\left(\mathcal{
M}_{2n+1},\xi\right)$ of an odd dimensional manifold and a contact
structure $\xi \, \stackrel{\mathcal{P}}{\longrightarrow}\,
\mathcal{M}_{2n+1}$.
\end{definizione}
Few relevant observations are in order in relation with the above
two definitions. The first is that the same contact structure can be
defined by several different contact forms $\alpha$,
$\alpha^\prime$,$\dots$. Indeed all multiples of a given contact
form $\alpha$ through a scalar, nowhere vanishing, function
$\lambda\, : \, \mathcal{ M}_{2n+1}\, \to \, \mathbb{R}$ define the
same contact structure. Secondly it is quite possible that the same
odd--dimensional manifold $\mathcal{ M}_{2n+1}$ can admit more that
one contact structure. The classification of these latter, modulo
trivial diffeomeorphisms, is an interesting and relevant
mathematical problem. It is therefore mandatory to single out the
notion of \textbf{contactomorphism}.
\begin{definizione}
Let $\left(\mathcal{ M},\xi\right)$ and $\left(\mathcal{
N},\chi\right)$ be two contact-manifolds and let:
\begin{equation}\label{diffeomorfus}
    \varphi \, : \, \mathcal{M} \, \longrightarrow \,\mathcal{N}
\end{equation}
be a diffeomorphism of the former on the latter manifold (obviously
$\mathcal{M}$ and $\mathcal{N}$ must have the same dimension in
order for $\varphi$ to possibly exist). Let $\alpha$ be a contact
form defining $\xi$ and let $\beta$ be a contact form defining
$\chi$. The considered diffeomorphism $\varphi $ is named a
\textbf{contactomorphism} if and only if:
\begin{equation}\label{pullabacco}
    \varphi^\star\left(\beta\right) \, = \, \lambda \, \, \alpha
\end{equation}
where $\varphi^\star$ is the pull-back map and
\begin{equation}\label{nowhere}
    \lambda \, : \, \mathcal{M} \, \longrightarrow \, \mathbb{R}
\end{equation}
is a nowhere vanishing real function on the contact manifold
$\mathcal{M}$. If a contactomorphism exists between them, the two
considered contact manifolds are named \textbf{contactomorphic}.
\end{definizione}
In the above definition the manifold $\mathcal{M}$ and $\mathcal{N}$
might be the same. In this case what we are actually considering is
the transformation by means of a diffeomorphism of a contact
structure into another one by means of a contactomorphism.
\begin{definizione}
Given two contact structures $\xi$ and $\chi$ on the same manifold
$\mathcal{M}_{2n+1}$ they are to be identified as the same if there
exists a \textit{contactomorphism} that maps one into the other.
\end{definizione}
In conclusion the relevant mathematical problem is that of
classifying contact structures on $\mathcal{M}_{2n+1}$ modulo
contactomorphisms.
\subsection{Integrability and Frobenius Theorem}
We shall not dwell on the details of Frobenius theorem about
integrability but rather just sketch the basic concepts underlying its
formulation. One just begins with the observation that every vector
field $\mathbf{X}$ on a manifold $\mathcal{M}$ of whatever
dimension defines its own integral curves $\mathcal{I}_\mathbf{X}$,
namely those curves that at any of their points admit the local
value of the vector field $\mathbf{X}$ as tangent vector. Since any
point $p\in \mathcal{M}$ lies on some integral curve
$\mathcal{I}_\mathbf{X}$, we are guaranteed that a single vector
field induces a \textit{foliation} of the manifold $\mathcal{M}$
into one-dimensional submanifolds. It is more tricky to establish
whether a sub-bundle of the tangent bundle $\mathcal{E}
\longrightarrow \mathcal{M}$ of rank  $r > 1$ induces or not a
foliation of $\mathcal{M}$. In this case, to say it in a
not-completely rigorous, yet intuitive and qualitatively correct
way, by \textit{foliation} we mean the covering of the manifold with
a family of \textit{leaves}, namely of disjoint sub-manifolds diffeomorphic
among themselves,  $\mathcal{F}_{\pmb{\nu}} \subset \mathcal{M}$ of
dimension equal to the rank $r$ of the sub-bundle $\mathcal{E}$,
each of which can be described, in a local patch, as the level set hypersurface  for $D-r$
functions $u_k(p)$ ($k=1,\dots,D-r$), and is spanned, in the same patch, by a set of $r$ functions $v_i(p)$, $i=1,\dots, r$, that originate from the
integration of a basis of sections $\mathbf{X}_i$ of the sub-bundle
$\mathcal{E} \longrightarrow \mathcal{M}$.
\begin{eqnarray}\label{sfogliatellanapoletana}
    \mathcal{F}_{\pmb{\nu}} & = & \left\{p \, \in \mathcal{M} \,
    \mid \, u_k(p) \, = \,\nu_k \right\} \quad ; \quad \pmb{\nu}\,
    \equiv\, \{\nu_1, \dots \nu_{D-r}\} \quad ; \quad \nu_k\, = \,
    \textit{real constants}\nonumber\\
    \nabla v_i(p) &=& \mathbf{X}_i\mid_p\,\,,\,\,\,\,i=1,\dots, r\,.
\end{eqnarray}
When the above situation is realized one says that the sub-bundle
$\mathcal{E} \longrightarrow \mathcal{M}$ is \textbf{integrable} and is also referred to as an \textbf{involutive distribution}.
\par
\textbf{Frobenius theorem} establishes a necessary sufficient condition for
such integrability.
\begin{teorema} Let $\mathcal{M}$ be a manifold and $\mathcal{E}
\longrightarrow \mathcal{M}$ a sub-bundle of its tangent bundle of
rank  $r > 1$. The necessary and sufficient condition for
$\mathcal{E}$ to be integrable is that:
\begin{equation}\label{frobeniata}
    \forall\, \mathbf{X},\mathbf{Y}  \in  \Gamma[\mathcal{E},\mathcal{M}] \quad :
    \quad
    \, \left[\mathbf{X},\mathbf{Y} \right] \, \in \, \Gamma[\mathcal{E},\mathcal{M}]
\end{equation}
\end{teorema}
In the case where $\mathcal{E} \longrightarrow \mathcal{M}$ is an
hyperplane bundle defined by a 1-form $\omega$ Frobenius
integrability condition can also be formulated as:
\begin{equation}\label{frombolino}
    \omega \, \wedge \, d\omega \, = \, 0\,.
\end{equation}
This shows that a contact structure defined by a contact form is the
exact opposite of an integrable sub-bundle. Indeed we can show that
it corresponds to maximal non-integrability.
\subsection{Isotropic submanifolds of a contact manifold and non integrability}
We begin with the following
\begin{definizione}
Let $\left(\mathcal{M}_{2n+1}\, , \, \xi\right)$ be a contact
manifold and $\mathcal{L}\subset\mathcal{M}_{2n+1}$ a submanifold.
Consider the tangent bundle of such a submanifold
$\mathcal{TL}\,\stackrel{\pi_{\tau}}{\longrightarrow}\, \mathcal{L}$
and the contact structure bundle $\xi
\stackrel{\pi_\xi}{\longrightarrow}\, \mathcal{M}$. The submanifold
$\mathcal{L}$ is named \textbf{isotropic} if and only if
\begin{equation}\label{isotropicamente}
    \forall p \in \mathcal{L} \quad  : \quad \pi_{\tau}^{-1}(p) \,
    \subset \, \pi_{\xi}^{-1}(p)
\end{equation}
Equivalently, if the contact structure $\xi$ is defined by the
contact-form $\alpha$, the sub-manifold $\mathcal{L}$ is
\textbf{isotropic} if any vector field $\mathbf{X}$ tangent to
$\mathcal{L}$,  is also in $\text{ker}\,\alpha$:
\begin{equation}\label{cundoiso}
\mathbf{X}\in \Gamma\left[\mathcal{TL},\mathcal{L}\right] \quad
\Rightarrow \quad \alpha(\mathbf{X}) \, = \, 0
\end{equation}
\end{definizione}
We  introduce the additional definition
\begin{definizione}
Let $\left(\mathcal{M}_{2n+1}\, , \, \xi\right)$ be a contact
manifold and $\widetilde{\mathcal{M}}_{2m+1} \subset \mathcal{M}_{2n+1}$
an odd dimensional submanifold of codimension $2(n-m)\geq 0$. Let
$\alpha$ be the contact one form defining the contact structure
$\xi$ and $\iota$ the inclusion map:
\begin{equation}\label{includendoiota}
    \iota \quad : \quad \widetilde{\mathcal{M}}_{2m+1} \,
    \longrightarrow \, \mathcal{M}_{2n+1}
\end{equation}
Then $\left(\widetilde{\mathcal{M}}_{2m+1},\chi\right)$ is named a
\textbf{contact submanifold} of $\left(\mathcal{M}_{2n+1}\, , \,
\xi\right)$ if the contact structure $\chi$ on
$\widetilde{\mathcal{M}}_{2m+1}$ is defined by the \textbf{contact-form}
$\iota^\star \, \alpha$, in other words if:
\begin{equation}\label{cariota}
    \chi \, = \, {\mathrm{ker}} \, \iota^\star\,\alpha
\end{equation}
\end{definizione}
The main reason why contact geometry is relevant for chaotic flows
in fluid dynamics streams from the following \cite{contatoregeiges}
\begin{teorema}
\label{barbagianni} Let $\left(\mathcal{M}_{2n+1},\xi\right)$ be a
contact manifold in $2n+1$-dimensions and
$\mathcal{L}\subset\mathcal{M}_{2n+1}$ an isotropic submanifold.
Then $\mathrm{dim}\, \mathcal{L} \leq n$.
\end{teorema}
In order to prove theorem \ref{barbagianni} we need first the
following
\begin{lemma}
Let  $\left(\mathcal{M}_{2n+1},\xi\right)$ be a contact manifold
whose contact structure $\xi$ is defined as  $\mathrm{ker}
\,\alpha$, in terms of the contact 1-form $\alpha$. Because of the
defining condition $\ref{latisanaditiglio}$ it follows that
$\mathrm{d}\alpha\mid_\xi \neq 0$ and for every point $p\in
\mathcal{M}_{2n+1}$ the $2n$-dimensional fibre $\xi_p \subset T_p
\,\mathcal{M}_{2n+1}$ is a vector-space equipped with a
skew-symmetric 2-form of maximal rank (no-zero eigenvalues) exactly
provided by the restriction to $\xi_p$ of $\mathrm{d}\alpha$
\textit{i.e.} $\mathrm{d}\alpha\mid_{\xi_p}$. Hence the contact
structure is a \textit{symplectic bundle} with respect to the 2-form
$\mathrm{d}\alpha\mid_\xi$.
\end{lemma}
\begin{proofteo}
{\rm In order to prove the theorem, consider the inclusion map:
\begin{equation}\label{legendro}
    \iota \quad : \quad \mathcal{L} \, \longrightarrow \, \mathcal{M}_{2n+1}
\end{equation}
and consider the pull-back of the contact form on the isotropic
manifold. By definition of isotropy $\iota^\star\alpha = 0$. Hence
we have also $\iota^\star \,\mathrm{d}\alpha = 0$. At each point
$p\in \mathcal{L}$, the tangent space $\mathcal{T}_p \mathcal{L}$ is
a subspace of the symplectic space $\xi_p$ on which the symplectic
2-form vanishes $\mathrm{d}\alpha\mid_{\xi_p}$. From elementary
linear algebra it follows that such a subspace has at most one-half
of the dimension of $\xi_p$. Indeed it suffices to put the skew
2-form in canonical form:
\begin{equation}
    \left(\begin{array}{c|c}
    \mathbf{0}_{n\times n}& \mathbf{1}_{n\times n}\\
    \hline
    -\mathbf{1}_{n\times n} &\mathbf{0}_{n\times n} \\
    \end{array}\right)\nonumber\\
\end{equation}
and the statement becomes evident. This proves the theorem
 $\blacksquare$.}
\end{proofteo}
\par
What are the consequences of this theorem? It states that if we have
a contact structure $\xi$, induced by a contact form $\alpha$, then
we can exclude a foliation of the contact manifold into
hypersurfaces $\Sigma_h \subset \mathcal{M}_{2n+1}$ of codimension
1:
\begin{equation}\label{sfoliazione}
  \mathcal{M}_{2n+1}\, \backsimeq \, \Sigma_h \times \mathbb{R}_h
\end{equation}
such that for each $h\in \mathbb{R}$ the  tangent bundle of
$\Sigma_h$ is comprised within the contact structure. Indeed if that
happened each leave $\Sigma_h$ of the foliation would be an
isotropic submanifold of dimension $2\times n$ which is what the
theorem forbids.
\begin{definizione}
An isotropic submanifold $\mathcal{L} \subset \mathcal{M}_{2n+1}$ of
maximal possible dimension, namely $n$, of a contact manifold in
dimensions $2n+1$, is named a \textbf{Legendrian submanifold}.
\end{definizione}
Furthermore
\begin{definizione}\label{ribatriestina} Associated with a contact form $\alpha$ one has
the so called \textbf{Reeb vector field}  $\mathbf{R}_\alpha$,
defined by the two conditions:
\begin{eqnarray}\label{Ribbo}
&&\alpha\left(\mathbf{R}_\alpha\right) \, = \, \lambda(\mathbf{x})
\quad =
 \quad
 \text{nowhere vanishing function on $\mathcal{M}_{2n+1}$}\nonumber\\
 &&\forall \mathbf{X}\in \Gamma\left[\mathcal{TM}_{2n+1},\mathcal{M}_{2n+1}\right] \quad :
 \quad
 \mathrm{d}\alpha\left(\mathbf{R}_\alpha,\mathbf{X}\right)\, = \,
 0
\end{eqnarray}
\end{definizione}
If the contact manifold $\mathcal{M}_{2n+1}$ is equipped with a
Riemannian metric $g$, then the contact 1-form $\alpha$ and its Reeb
field $\mathbf{R}_\alpha$ are related one to the other by the
raising and lowering of indices. Suppose that we start from the Reeb
field:
\begin{equation}\label{ribbone}
    \mathbf{R} \, = \, R^\mu \,
    \frac{\partial}{\partial x^\mu}
\end{equation}
The corresponding $\alpha$ is obtained by setting:
\begin{equation}\label{contactribbo}
    \alpha \, = \, \Omega^{[\mathbf{R}]} \,  \equiv \, g_{\mu\nu} R^\mu \,
    dx^\nu
\end{equation}
and the contact structure condition (\ref{latisanaditiglio}) is
turned into the following condition on the Reeb field components:
\begin{equation}\label{gridolino}
    \epsilon^{\lambda \mu_1\nu_1\mu_2\nu_2\dots\mu_n\nu_m} \,
    R_\lambda \, \partial_{\mu_1}\,R_{\nu_1}
    \,\partial_{\mu_2}\,R_{\nu_2}\, \dots\,
    \partial_{\mu_n}\,R_{\nu_n}\, \neq \, 0 \quad \text{nowhere vanishes}
\end{equation}
On the contrary, if one begins with the contact form $\alpha$, the
components of the Reeb field are obtained by setting:
\begin{equation}\label{crinolina}
    R_\alpha^\mu \, = \, g^{\mu\nu} \, \alpha_\nu
\end{equation}
Note that the nowhere vanishing function $\lambda$ mentioned in the
definition \ref{ribatriestina} is just the squared norm of the Reeb
field or of the contact form which coincide:
\begin{equation}\label{normaquadrata}
    \lambda \, =\, \parallel \mathbf{R}\parallel^2 \, = \, \parallel
    \Omega^{[\mathbf{R}]}\parallel^2 \, \equiv \, g_{\mu\nu} \, R^\mu \,
    R^\nu
\end{equation}
\subsection{Contact structures in $D=3$ and hydro-flows}
Let us now consider the case relevant for fluid dynamics, namely
that of three dimensional contact manifolds
$\left(\mathcal{M}_3\,\xi_\alpha\right)$, where, in the notation
$\xi_\alpha$, we mention the contact form $\alpha$ defining the
contact structure. The consequence of theorem \ref{barbagianni} is
that in such contact manifolds, the Legendrian submanifolds are all
1-dimensional, namely they are \textit{curves} or, as it is
customary to name them in the present context, \textit{knots}.
\par
Hence in three dimensions there are two kind of knots, the
\textbf{Legendrian knots} whose tangent vector belongs to
$\text{ker}\, \alpha$ and the \textbf{transverse knots} whose
tangent vector is parallel to the Reeb field at each point of the
trajectory.
\par
Furthermore in D=3 the condition (\ref{gridolino}) becomes:
\begin{equation}\label{urlonotturno}
    \epsilon^{\lambda\mu\nu}  \,R_\lambda \, \partial_\mu \, R_\nu
    \, \neq \, 0
\end{equation}
\paragraph{The standard contact structure on $\mathbb{R}^3$.} The
flat Euclidian space in three dimensions whose coordinates we denote
as $x,y,z$ is endowed with a standard contact structure that admits
the following contact form:
\begin{equation}\label{romildino}
    \alpha_s \, = \, \mathrm{d}z+ x \,\mathrm{d}y
\end{equation}
A picture of the local planes defining the contact structure
(\ref{romildino}) is shown in fig.\ref{berlucchino}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=150mm]{kontaktusR3.png}
\end{center}
\caption{\it  Schematic view of  the standard contact structure
$\mathbb{R}^3$} \label{berlucchino}
 \hskip 1cm \unitlength=1.1mm
\end{figure}
\subsection{Relation with Beltrami vector fields}
As we see, the main reason to introduce the concept of contact form is that, by doing so, one liberates the notion of a vector field capable
of generating chaotic trajectories from the use of any  metric
structure. A vector field $U$ is potentially interesting for chaotic
regimes if it is a Reeb  field for at least one contact form
$\alpha$. In this way the mathematical theorems about the
classification of contact structures modulo diffeomorphisms
(theorems that are metric-free and of topological nature) provide
new global methods to capture the topology of hydro-flows.
\par
Instead if we work in a Riemaniann manifold endowed with a metric
$(\mathcal{M},g)$ we can always invert the procedure and define the
contact form $\alpha$ that can admit $\mathrm{U}$ as a Reeb vector
field by identifying
\begin{equation}\label{gominato}
    \alpha \, = \, \Omega^{[\mathrm{U}]}
\end{equation}
In this way the first of the two conditions (\ref{Ribbo}) is
automatically satisfied: ${\rm i}_\mathrm{U}\Omega^{[\mathrm{U}]} \,
= \, \parallel \mathrm{U}\parallel^2 \, > 0$. It remains to be seen
whether $\Omega^{[\mathrm{U}]}$ is indeed a contact form, namely
whether $\Omega^{[\mathrm{U}]} \, \wedge \, \mathrm{d}
\Omega^{[\mathrm{U}]} \, \ne \,0$ and whether the second condition
${\rm i}_\mathrm{U} \, \mathrm{d} \Omega^{[\mathrm{U}]} \, = \, 0$
is also satisfied. Both conditions are automatically fulfilled if
$\mathrm{U}$ is a Beltrami field, namely if it is an eigenstate of
the operator $\star_g \, \mathrm{d}$ as advocated in
eq.(\ref{Belatramus}). Indeed the implication ${\rm i}_\mathrm{U} \,
\mathrm{d} \Omega^{[\mathrm{U}]} \, = \, 0$ of Beltrami equation was
shown  in eq. (\ref{cinesinotulipano}), while from the Beltrami
condition it also follows:
\begin{eqnarray}\label{cirimellaG}
 \Omega^{[\mathrm{U}]} \, \wedge \, \mathrm{d} \Omega^{[\mathrm{U}]}
 & = &\Omega^{[\mathrm{U}]} \, \wedge \, \star_g \Omega^{[\mathrm{U}]}
 \, = \, \parallel \mathrm{U}\parallel^2 \, \mathrm{Vol}  \, \ne \, 0 \nonumber\\
 \mathrm{Vol} & \equiv & \frac{1}{3!} \,
 \times \, \epsilon_{ijk} \, dx^i \, \wedge \, dx^j \, \wedge \, dx^k
\end{eqnarray}
In this way the conceptual circle closes and we see that all
Beltrami vector fields can be regarded as Reeb  fields for a
bona-fide contact form. Since the same contact structure (in the
topological sense) can be described by different contact forms, once
Beltrami fields have been classified it remains the task to discover
how many inequivalent contact structures they actually describe. Yet
it is reasonable to assume that every contact structure has a
contact form representative that is derived from a Beltrami Reeb
field. Indeed a precise correspondence is established by a theorem
proved in \cite{Ghrist2007}:
\begin{teorema}
\label{ghristheo} Any rotational Beltrami vector field on a
Riemaniann $3$-manifold is a Reeb  field for some contact form.
Conversely any Reeb field associated to a contact form on a
$3$-manifold is a rotational Beltrami field for some Riemaniann
metric. Rotational Beltrami field means an eigenfunction of the
$\star_g \mathrm{d}$ operator corresponding to a non vanishing
eigenvalue $\lambda$.
\end{teorema}
\subsection{Darboux's theorem}
We finally mention, without providing its proof that can be found in
\cite{contatoregeiges}, a  classical theorem named after Darboux,
which shows that the standard contact structure on $\mathbb{R}^3$
displayed in eq.(\ref{romildino}) and graphically shown in
fig.\ref{berlucchino} is not just a choice, rather it corresponds to
the  canonical local form of any contact structure on any contact
manifold.
\begin{teorema}
\label{darbuso} Let $\left(\mathcal{M}_{2n+1},\xi\right)$ be an
$(2n+1)$-dimensional contact manifold and $\alpha$ a contact
$1$-form defining $\xi = \text{ker}\,\alpha$. Let $p\in
\mathcal{M}_{2n+1}$ be any point of the manifold and $U \subset
\mathcal{M}_{2n+1}$ an open neighborhood of $p$. Then we can always
find a local homomorphism: $\varphi \, : \, U \, \to \,
\mathbb{R}^{2n+1}$ such that, naming
$\{y_0,x_i,y_i\}$,$(i=1,\dots,n)$ the coordinates on $\varphi(U)
\subset \mathbb{R}^{2n+1}$ we obtain:
\begin{equation}\label{localU}
    \alpha\mid_{U} \, = \, dy_0 \, + \,\sum_{i=1}^n \,x_i\,dy_i
\end{equation}
\end{teorema}
In the case $n=1$ eq.(\ref{localU}) reproduces eq.(\ref{romildino}).
Hence for all three-dimensional contact manifolds $\mathcal{M}_3$
that in (\ref{romildino}) is the universal local form of the contact
$1$-form $\alpha$.
\section{$\mathfrak{b}$-Contact Geometry and Singular Beltrami Fields}
\label{bmanstory}
As we emphasized in the introduction, the main difficulty in solving
NS or Euler equations comes from the non-linearity of the transport
term which forbids the generic linear superposition of solutions, a
limited superposition being possible, within the landscape approach
to be discussed in chapter \ref{panoramica}, with Beltrami fields
belonging to the same spherical layer. As we stressed, Beltrami
fields are essential, via their relation with contact structures, in
order to create the possibility of chaotic streamlines at small
scales, yet they are defined on compact manifolds without boundary,
in particular on torii, while the geometry of physical systems of
relevance for applications is certainly  not that of torii, rather
that of finite portions of $\mathbb{R}^3$ delimited by boundaries,
like finite 3D cylinders. Furthermore at larger scales, the fluids
of interest for applications should present a non chaotic behavior
similar to that of the Poiseuille flow (in section \ref{pescheria}
see eq.s (\ref{pescetto}-\ref{soluso2}) and the related figure
\ref{caledonia}) or the shear vortex (in section \ref{genbeltraflow}
see eq.(\ref{caluno}) and the related fig.\ref{alieno}). How could
we try to reconcile the two conflicting needs? A new window of
opportunity opens up with the relatively new set up of Beltrami
fields in $\mathfrak{b}$-manifolds, which  can be viewed as compact
manifolds with boundaries. In this section we make a short review of
this new approach which, as already stressed, we desire to combine
with our group theoretical classification of Beltrami fields.
Essentially we collect the main definitions and concepts developed
in particular by Victor Guillemin, Eva Miranda, Robert Cardona,
Daniel Peralta Salas and other collaborators in
\cite{cardone2019,miranda2021,trasversozero,unochiuseinteg,guglielminoconmiranda,pollosingolare},
having, as main goal, that of discussing the example of the
$\mathfrak{b}$-modified ABC flow\footnote{see section
\ref{abcperbambini} for the definition of ABC flows} presented in
\cite{miranda2021}. Such a discussion will be done, in view of the
underlying group theoretical structures, in section \ref{bABCfildo}.
\par
In order to introduce the $\mathfrak{b}$-generalization of contact
manifolds we have first to set the stage by recalling essential
facts and definitions about symplectic manifolds Poisson structures
and transversality.
\subsection{Symplectic and Poisson Manifolds}
We begin with
\begin{definizione}
A symplectic manifold is a pair
$\left(\mathcal{SM}_{2n+2},\omega\right)$ of a smooth manifold
$\mathcal{SM}_{2n+2}$ in even dimension $2n+2$ and a $2$-form
$\omega$ which is closed and non degenerate of maximal rank:
\begin{equation}\label{simplicioformo}
    \mathrm{d}\omega \, = \, 0 \quad ; \quad
    \omega\wedge\omega\wedge \dots \wedge \omega \, \neq 0 \quad
    \text{everywhere on $\mathcal{SM}_{2n+2}$}
\end{equation}
\end{definizione}
On a symplectic manifold we have a naturally defined antisymmetric
quadratic form on the space of sections of the tangent bundle,
\textit{i.e.} on the vector fields:
\begin{eqnarray}\label{tardicchio}
    \omega &: & \Gamma\left[\mathcal{TSM}_{2n+2}\, , \,
    \mathcal{SM}_{2n+2}\right] \times \Gamma\left[\mathcal{TSM}_{2n+2}\, , \,
    \mathcal{SM}_{2n+2}\right] \, \longrightarrow \,
    C^{(\infty)}\left(\mathcal{SM}_{2n+2}\right)\nonumber\\
   \forall X,Y\, &\in & \Gamma\left[\mathcal{TSM}_{2n+2}\, , \,
    \mathcal{SM}_{2n+2}\right] \quad , \quad  \omega\left(X,Y\right) \,
    \in\,
    C^{(\infty)}\left(\mathcal{SM}_{2n+2}\right)
\end{eqnarray}
Poisson manifolds are instead defined as follows.
\begin{definizione}
\label{poissone} A Poisson manifold
$\left(\mathcal{PM}_m,\{,\}\right)$ is the pair of a smooth manifold
$\mathcal{PM}_m$ of dimension $m$ and a Poisson bracket $\{,\}$
which is a binary operation on the space of smooth functions on the
manifold:
\begin{equation}\label{pesceparente}
    \{,\} \quad : \quad C^{(\infty)} \left(\mathcal{PM}_m\right)
    \, \times \, C^{(\infty)} \left(\mathcal{PM}_m\right)\,
    \longrightarrow \, C^{(\infty)} \left(\mathcal{PM}_m\right)
\end{equation}
satisfying the following three properties:
\begin{description}
  \item[1)] Antisymmetry $\quad \{f\, ,\, g\} \, = \, - \, \{g\, ,\, f\}
  $, $\quad\forall f,g \in C^{(\infty)} \left(\mathcal{PM}_m\right)$
  \item[2)] Jacobi Identity $\quad \{f\, , \, \{g \, ,\, h\}\}+\{g\, , \, \{h \, ,\, f\}\}+\{h\, , \, \{f \, ,\, g\}\}\, = \,
  0$, $\quad \forall f,g,h \in C^{(\infty)} \left(\mathcal{PM}_m\right)$
  \item[3)] Leibniz rule $\quad \{f\, , \, g.h\} \, = \, \{f\, , \,
  g\} \, h \, + \, g \,\{f\, , \,
  h\}$,  $\quad \forall f,g,h \in C^{(\infty)} \left(\mathcal{PM}_m\right)$
\end{description}
\end{definizione}
The first two properties mentioned in the definition \ref{poissone}
guarantee that the space of functions on the Poisson manifold
becomes a Lie algebra once equipped with the Poisson bracket. On the
other end the third property implies that to each function $f \in
C^{(\infty)}$ the Poisson bracket associates a derivation of the
commutative algebra of functions on the manifold, namely, by
definition a vector field $\mathbf{X}_f$, named the
\textbf{hamiltonian vector field} of $f$.
\par
Locally, in any coordinate patch $\{x_1,\dots,x_j\}$, the Poisson
bracket takes the following form:
\begin{equation}\label{localedelpesce}
    \{f,g\} \, = \,\pi^{ij}(x)\,\frac{\partial f}{\partial x^i}\, \frac{\partial g}{\partial
    x^j} \quad ; \quad \pi^{,j}(x) \, = \, - \, \pi^{ji}(x)
\end{equation}
where the controvariant antisymmetric tensor $\pi^{ij}(x)$ is
usually called a \textbf{bivector}. The hamiltonian vector field
$\mathbf{X}_f$ is then easily identified:
\begin{equation}\label{carneadehamilt}
    \mathbf{X}_f \, = \, \pi^{ij}\partial_i f \, \partial_j
\end{equation}
Let us now suppose that the dimension of the Poisson manifold is
even $m=2n+2$ and that the bivector $\pi^{ji}(x)$ is an everywhere
invertible matrix. Setting:
\begin{equation}\label{grundlagen}
    \omega \, = \, \pi^{-1}_{k\ell} dx^k \wedge dx^\ell
\end{equation}
we obtain a symplectic 2-form of maximal rank which is closed as a
consequence of the Jacobi identities satisfied by the bivector. In
this way the Poisson manifold is recognized to be a symplectic
manifold. In particular we can set:
\begin{equation}\label{agniziono}
    \{f,g\} \, = \, \omega\left(\mathbf{X}_f,\mathbf{X}_g\right)
\end{equation}
\begin{definizione}
\label{liuvillocampillo} Let
$\left(\mathcal{SM}_{2n+2},\omega\right)$ be a symplectic manifold.
A \text{Liouville vector field} is a vector field that leaves the
symplectic form $\omega$ invariant, namely:
\begin{equation}\label{taleggio}
  \omega\, = \,  \mathcal{L}_X\omega \, \equiv \, i_X\,
  \mathrm{d}\omega \, + \, \mathrm{d}\,\left( i_X\omega\right)
\end{equation}
where $\mathcal{L}_X$ denotes the Lie derivative along the vector
field $X$ and $i_X$ the contraction of a form with the same vector field.
\end{definizione}
\subsection{Transversality}
\begin{definizione}
\label{trasversimanif} Let $\Upsilon$ be a manifold and
$\mathcal{M}\subset \Upsilon$, $\mathcal{N}\subset \Upsilon$ be two
submanifolds. The two manifolds $\mathcal{M}$ and $\mathcal{N}$ are
\textbf{transverse} in $\Upsilon$ if:
\begin{equation}\label{carnevaleviareggio}
    \forall p \, \in \,\mathcal{M}\bigcap\mathcal{N} \quad:\quad
    T_p\mathcal{M} \, + \,T_p\mathcal{N} \, = \, T_p\Upsilon
\end{equation}
\end{definizione}
Obviously eq.(\ref{carnevaleviareggio}) implies that the sum of the
dimensions of $M$ and $N$ equals the dimension of the ambient
manifold $\Upsilon$.
\par
The concept of transversality can be applied to maps among manifolds
according with the following definition.
\begin{definizione}
\label{trasversmap} Let X and Y be two manifolds and let:
\begin{equation}\label{smuttomappo}
    f \, : \quad  X \, \longrightarrow \, Y
\end{equation}
be a smooth map. Let $Z\subset Y$ be a submanifold of the target
manifold $Y$. We say that $f$ is transverse to $Z$ if:
\begin{equation}\label{traversinaferroviaria}
    \forall q\, \in \, f^{-1}\left(Z\right) \quad : \quad T_{f(q)}Y
    \, = \, T_{f(q)}Z + f_\star\left(T_a X\right)
\end{equation}
where $f_\star$ is the push-forward map.
\end{definizione}
Finally the concept of transversality can be applied to vector
fields and hypersurfaces.
\begin{definizione}
\label{paninoconricotta} Let $\mathcal{M}$ be a manifold and
$\Sigma\subset \mathcal{M}$ a hypersurface, namely a codimension one
submanifold. We say that $\Sigma$ is transverse to the vector field
$\mathbf{V}$ if:
\begin{equation}\label{trasversvettor}
    \forall q\, \in \, \Sigma \quad : \quad T_q\mathcal{M}\, =
    \,T_q\Sigma\, + \,\mathbf{V}_q
\end{equation}
\end{definizione}
A notable example is provided by the potential flow analyzed in
section \ref{potenzialflusso}. Any level surface of the potential:
\begin{equation}\label{lamentoso}
    \Sigma_h\, = \, \left\{ \mathbf{x}\in \mathbb{R}^3\mid
    \phi(\mathbf{x}) \, = \, h \right\}
\end{equation}
is transverse to gradient vector field $\mathbf{X}_\phi \, = \,
\partial_j\phi(\mathbf{x})\,g^{ij} \partial_j$.
\subsection{Relation between symplectic and contact manifolds}
Let us consider a symplectic manifold $\left(
\mathcal{SM}_{2n+2},\omega\right)$ and let us assume that it admits
at least one Liouville vector field $\mathbf{L}$. Let moreover
$\Sigma_{\mathbf{L}}\subset \mathcal{SM}_{2n+2} $ be a hypersurface
transverse to the Liouville vector field $\mathbf{L}$ according to
the definition \ref{paninoconricotta}. Then we realize that
$\Sigma_{\mathbf{L}}$ is a contact manifold with contact form:
\begin{equation}\label{robedamatti}
    \alpha \, = \,i_{\mathbf{L}}\omega
\end{equation}
If $\Sigma_{\mathbf{L}}$ is transverse to $\mathbf{L}$ the form
$\alpha$ vanishes on $\mathbf{L}$ and no-where vanishes on
$T\Sigma_{\mathbf{L}}$. To verify that it is indeed a contact form
we just have to compute:
\begin{eqnarray}\label{gelso}
    &\null&\alpha\wedge \underbrace{\mathrm{d}\alpha \wedge \dots \wedge
    \mathrm{d}\alpha}_{n-\text{times}} \, = \, i_\mathbf{L}\omega\wedge\underbrace{\mathrm{d} i_\mathbf{L}\omega \wedge \dots \wedge
    \mathrm{d} i_\mathbf{L}\omega}_{n-\text{times}}\,=\,i_\mathbf{L}\omega\wedge\underbrace{\omega \wedge \dots \wedge
    \omega}_{n-\text{times}}\nonumber\\&& = \, \frac{1}{n+1} i_\mathbf{L}\left(\underbrace{\omega \wedge \dots \wedge
    \omega}_{(n+1)-\text{times}}
    \right)\, = \, \text{Vol}_{\Sigma_\mathbf{L}}
\end{eqnarray}
The last equation is true because $\omega^{n+1}$ is the volume form
of the ambient symplectic manifold and the hypersurface
$\Sigma_\mathbf{L}$ is by hypothesis transverse to the Liouville
vector field.
\par
Conversely given a contact manifold
$\left(\mathcal{M}_{2n+1},\xi\right)$ with contact form $\alpha$ and
Reeb field $\mathbf{R}$, any hypersurface $\Sigma \subset
\mathcal{M}_{2n+1}$ which is transverse to the Reeb field
$\mathbf{R}$ automatically acquires the structure of a symplectic
manifold with symplectic form $\widetilde{\omega}\, = \,
d\alpha\mid_\Sigma$.
\par
Hence we can have odd-dimensional contact manifolds that sit in
between two symplectic manifolds of adjacent dimensions as shown in
the following diagram:
\begin{equation}\label{diagrammus}
\begin{array}{ccccc}
    \left(\mathcal{SM}_{2n},\widetilde{\omega}=d\alpha\right)
    &\stackrel{\iota}{\hookrightarrow}
    &\left(\mathcal{M}_{2n+1},\alpha=i_\mathbf{L}\omega\right)
    &\stackrel{\iota}{\hookrightarrow}
    & \left(\mathcal{SM}_{2n+2},\omega\right)\\
    \Downarrow &\null& \Downarrow &\null & \Downarrow\\
    \text{symplectic}&\null& \text{contact}& \null &
    \text{symplectic}\\
\text{transverse to Reeb field}&\null& \text{transverse to Liouville
field}& \null &
    \null\\
    \end{array}
\end{equation}
The scheme described in eq.(\ref{diagrammus}) reminds that occurring
with Sasaki manifolds that sit in between two K\"ahler manifolds
which, indeed, are special instances of symplectic manifold, the
symplectic form being the K\"ahler 2-form.
\subsection{$\mathfrak{b}$-Manifolds}
Having clarified the above concepts and definitions we come to our
main goal that is the definition of $\mathfrak{b}$-manifolds.
Following \cite{trasversozero,cardone2019} we set:
\begin{definizione}
\label{bmanif} A $\mathfrak{b}$-manifold is a pair
$(\mathcal{M},\Sigma)$ where $\mathcal{M}$ is a differentiable
manifold and $\Sigma \subset \mathcal{M}$ is a hypersurface, namely
a submanifold of codimension one.
\end{definizione}
Given two $\mathfrak{b}$-manifolds $(\mathcal{M},\Sigma)$ and
$(\mathcal{N},\Pi)$ one defines as follows a smooth
$\mathfrak{b}$-map between them.
\begin{definizione}
A smooth map
\begin{equation}\label{mappas}
    f \quad : \quad \mathcal{M} \longrightarrow \mathcal{N}
\end{equation}
is a $\mathfrak{b}$-map:
\begin{equation}\label{bimappas}
    \null^\mathfrak{b}f \quad : \quad (\mathcal{M},\Sigma) \,\longrightarrow\,(\mathcal{N},\Pi)
\end{equation}
if $f$ is transverse to $\Pi$ and $f^{-1}(\Pi) \, = \, \Sigma$
\end{definizione}
With this setup one can re-establish all the basic ingredients of
differential geometry in the $\mathfrak{b}$-version. We begin with
vector fields.
\begin{definizione}
A $\mathfrak{b}$-vector field on $\mathfrak{b}$-manifold
$(\mathcal{M}_{m+1},\Sigma_m)$ is a vector field
$\null^{\mathfrak{b}}\mathbf{X}$ which is tangent to the
hypersurface $\Sigma_m$ in all $p\in \Sigma_m$
\end{definizione}
In an open neighborhood $U \subset \mathcal{M}_{m+1}$  that contains
the point $p\in \Sigma$ we can choose the coordinates in the
following way. Let $\sigma(x_0,x_1,\dots,x_{m})$ be the function
whose vanishing defines the surface $\Sigma_m$ in that neighborhood.
We can trade one of the standard coordinates $x_i$, say  $x_0$, for
the value $s=\sigma(x_0,\dots,x_{m+1})$ of the function, regarding
the remaining ones  $\mathbf{x}=\{x_1,\dots,x_m\}$ as coordinates on
the hypersurface $\Sigma$. Using such coordinate frame a vector
field parallel to the surface is of the form:
\begin{equation}\label{parallelone}
    \null^{\mathfrak{b}}\mathbf{X}\, =\, s
    X^0(s,\mathbf{x})\frac{\partial}{\partial s} + \sum_{i=1}^m
    X^i(s,\mathbf{x}) \frac{\partial}{\partial x^i}\,,
\end{equation}
where $X^0(s,\mathbf{x}),\,X^i(s,\mathbf{x})$ are smooth functions of $s,x_1,\dots, x_n$.
One can easily check that under standard commutation the
${\mathfrak{b}}$-vector fields form a Lie subalgebra of the Lie
algebra of vector fields. They can be considered the sections of a
new vector-bundle on $\mathcal{M}_{m+1}$ that we name the
$\mathfrak{b}$-tangent bundle:
$\null^{\mathfrak{b}}T\mathcal{M}_{m+1}\,\stackrel{\pi}{\longrightarrow}\,{\mathcal{M}}_{m+1}$.
This being established the road easily climbs down. We obtain the
the $\mathfrak{b}$-cotangent bundle by usual duality.
\par
In practice as shown in \cite{guglielminoconmiranda} the
$\mathfrak{b}$ de-Rham complex is structured as follows. A $k$-form
$\null^{\mathfrak{b}}\omega\,\in\,\null^{\mathfrak{b}}\Omega^k(\mathcal{M})$,
namely a section of the $k$-th external power of the cotangent
bundle $\null^{\mathfrak{b}}T^\star \mathcal{M}$ can always be
written as:
\begin{equation}\label{alfabetone}
\null^{\mathfrak{b}}\omega \, = \,    \frac{ds}{s}\wedge \alpha \, +
\, \beta \quad ; \quad \alpha\,\in\,\Omega^{k-1}(\mathcal{M})\quad ;
\quad
    \beta\,\in\,\Omega^{k}(\mathcal{M})\,.
\end{equation}
Furthermore in \cite{guglielminoconmiranda} it is stated and shown
that although $\alpha,\beta$ are not unique in the bulk of the
manifold they are unique at every point $p\, \in \, \Sigma$ on the
distinguished surface or boundary.
\par
This provides an algorithmic tool to perform the $b$-deformation of
any given Riemannian metric on a given manifold $\mathcal{M}$.
\par
Relevant to our goals is the $\mathfrak{b}$-generalization of the
definition of contact manifolds.
\begin{definizione} Let $(\mathcal{M},\Sigma)$ be a
$(2n+1)$-dimensional, $\mathfrak{b}$-manifold. A
$\mathfrak{b}$-contact structure is the kernel of a $b$-one-form
$\alpha\, \in \, \null^{\mathfrak{b}}T^\star\mathcal{M}$ that
satisfies the condition:
\begin{equation}\label{ravanone}
    \alpha \wedge \mathrm{d}\alpha \wedge \dots \wedge \mathrm{d}\alpha\, \neq \, 0
\end{equation}
In this case $\alpha$ is a $\mathfrak{b}$-contact form and $\xi \, =
\, \text{ker}\,\alpha$ is a $\mathfrak{b}$-contact structure.
\end{definizione}
As in the un-deformed case we can introduce the $\mathfrak{b}$-Reeb
field as that particular $\mathfrak{b}$-vector field $\mathbf{R}$
which satisfies the two conditions:
\begin{equation}\label{ribbetto}
    i_\mathbf{R}\cdot \mathrm{d}\alpha \, = \, 0 \quad ; \quad
    \alpha(\mathbf{R}) \, = \, 1
\end{equation}
As we are going to see in section \ref{bABCfildo}, the use of
$\mathfrak{b}$-deformations can introduce modified Beltrami fields
that are parallel to certain boundaries. The open deep question that
is touched upon and put into evidence in chapter \ref{dippo} is that
the choice of an allowed distinguished surface $\Sigma$ seems to
depend on the group structure of the Beltrami field one wants to
$\mathfrak{b}$-deform. Up to the knowledge of the authors this
aspect has not been discussed in the literature so far. It appears to be
a very momentous question worth an in depth investigation.
\chapter{Harmonic Analysis and the Algorithm}
\section{Beltrami equation and harmonic analysis}
In the present section which is largely based on a corresponding
section of \cite{Fre:2015mla} we stress that all the arguments
presented above have been instrumental to enlighten the role of
Beltrami vector fields from various viewpoints related with
hydrodynamics and lagrangian chaos. Let us now consider from a more
general point of view Beltrami equation (\ref{Belatramus}). The one
here at stake  is the case $p=1$ of an eigenvalue equation that can
be written in any $(2\, p + 1)$-dimensional Riemaniann manifold
$\left(\mathcal{M}_p \, , \, g\right)$, namely:
\begin{equation}\label{choucrut}
    \star_g  \mathrm{d}\omega^{(p)} \, = \, \lambda \, \omega^{(p)}
\end{equation}
The eigenfunctions of the $\star_g  \mathrm{d}$ operator are 1-forms
for $p=1$, namely in three-dimensions, but they are higher
differential forms in higher odd dimensions. Another particularly
interesting case is that of $7$-manifolds where the eigenfunctions
of $\star_g  \mathrm{d}$ are three-forms and can be related with a
$\mathrm{G_2}$-structure of the manifold
\cite{englert,miol168,Cerchiai:2018shs}. On the other hand the
relation encoded in theorem \ref{ghristheo} between
eq.(\ref{choucrut}) and contact structures, as they are defined in
current mathematical literature, is true only for $p=1$ and it is
lost for higher $p$. Indeed contact structures are always defined in
terms of a contact one-form and by the condition:
\begin{equation}\label{salsicciafresca}
    \alpha \, \wedge \, \underbrace{\mathrm{d}\alpha \,
    \wedge \,\mathrm{d}\alpha \, \dots \, \mathrm{d}\alpha}_{p-\mbox{times}} \, \ne \, 0
\end{equation}
Hence the problem of determining the spectrum and the eigenfunctions
of the operator $\star_g  \mathrm{d}\omega^{(p)}$  is a general one
and can be addressed in the same way in all odd-dimensions, yet its
relation with flows and contact-structures is peculiar to $d=3$ and
has not a general significance.  In any case it is absolutely  clear
that once the correspondence of theorem \ref{ghristheo} has been
established, the classification of Beltrami fields is reduced to a
classical problem of differential geometry whose solution can be
derived within a time honored framework which makes no reference to
trajectories and contact structures.
\par
The framework we refer to is that of \textit{harmonic analysis} on
compact Riemaniann manifolds $\left( \mathcal{M},g\right)$  and its
application to the spectral analysis of Laplace-Beltrami  operators
(for reviews see the book \cite{castdauriafre} and the articles
\cite{univer,spectfer,Fabbri:1999mk,Fre1999xp,gualto}). As
thoroughly discussed in the quoted references there are, on a
Riemann manifold $\left( \mathcal{M},g\right)$, several invariant
differential operators, generically named Laplace-Beltrami some of
which are of second order, some other of first order. They act on
the sections of vector bundles $E \rightarrow \mathcal{M}$  of
different rank, for instance the tangent bundle, the bundle of
$p$-forms, the bundle of symmetric two tensors, the spinor bundle
etc. Among the first order operators the most important ones are the
Dirac operator acting on sections of the spinor bundle and the
$\star_g \mathrm{d}$-operator acting on $p$-forms in a $(2 \, p
+1)$-dimensional manifold. The spectrum of all Laplace-Beltrami
operators is sensitive both to the topology and to the metric of the
underlying manifold. Each eigenspace is organized into irreducible
representations of the isometry group $\mathrm{G}$ of the metric $g$
and the eigenfunctions assigned to a particular representation are
generically named \textit{harmonics}.
\par
Here comes an important distinction in relation with the nature of
the group $\mathrm{G}$. If $\mathrm{G}$  is a Lie group and if the
manifold $\mathcal{M}$ is homogeneous under its action, than
$\mathcal{M} \sim \mathrm{G/H}$ where $\mathrm{H}\subset \mathrm{G}$
is the stability subgroup of some reference point $p_0 \, \in \,
\mathcal{M}$. In this case harmonic analysis reduces completely to
group-theory and the spectrum of any Laplace-Beltrami operator can
be derived in pure algebraic terms without ever using any
differential operations. In the case $\mathrm{G}$ is not a Lie group
and/or $\mathcal{M}$ is not homogeneous under its action, then
matters become more complicated and \textit{ad hoc} techniques have
to be utilized case by case to analyze the spectrum of invariant
operators.

\subsection{Harmonic analysis on the $\mathrm{T^3}$ torus and the Universal Classifying Group}
\label{introT3} The reasons to compactify Arnold-Beltrami flows on a
$T^3$ have already been discussed and we do not resume the issue. We
just observe that $\mathbb{R}^3$ is a non-compact coset manifold so
that harmonic analysis over $\mathbb{R}^3$ is a complicated matter
of functional analysis. After compactification, namely after
imposing periodic boundary conditions, things drastically simplify.
\par
Firstly, as we explained above  the compactification is obtained by
quotienting $\mathbb{R}^3$ with respect to a discrete subgroup of
the translation group which constitutes a lattice (see
eq.(\ref{metricT3})).
\par Secondly we implement the programme of
harmonic analysis by presenting a general algorithm to construct
solutions of the Beltrami equation which utilizes as main ingredient
the orbits under the action of the point group
$\mathfrak{P}_\Lambda$ of three-vectors in the momentum lattice
$^\star\Lambda$ that is just the dual of the lattice $\Lambda$. In
the language of crystallography the point group  is just the
discrete subgroup $\mathfrak{P}_\Lambda \subset \mathrm{SO(3)}$ of
the rotation group which maps the lattice $\Lambda$ and its dual
$^\star\Lambda$ into themselves:
\begin{equation}\label{puntarellona}
    \mathfrak{P}_\Lambda \, \Lambda \, = \, \Lambda \quad ; \quad \mathfrak{P}_\Lambda \, ^\star\Lambda\, = \, ^\star\Lambda
\end{equation}
In the case of the cubic lattice, that is the main example studied
in paper \cite{Fre:2015mla} we have  $\mathrm{G_{cubic}}\, = \,
\mathrm{O_{24}}$ where $\mathrm{O_{24}} \sim \mathrm{S_4}$ is the
proper octahedral group of order $|\mathrm{O_{24}}|\, = \, 24$. In
the case of the hexagonal lattice which was only briefly touched
upon in \cite{Fre:2015mla} and which  instead we analyze in depth in
the present work, the point group is the dihedral group
$\mathrm{D_6}$ of order $|\mathrm{D_{6}}|\, = \, 12$.
\par
Thirdly, as it was originally  conceived and introduced for the
first time in \cite{Fre:2015mla},  a general argument, inspired by
the logic that crystallographers used to derive and classify space
groups, leads to introduce a large finite group
$\mathfrak{GU}_\Lambda$, named by the authors of \cite{Fre:2015mla}
the \textit{Universal Classifying Group for the Lattice $\Lambda$},
made out of discretized rotations and translations that are defined
by the structure of $\Lambda$. All eigenfunctions of the $\star_g
\mathrm{d}$-operator can be organized into a finite number of
classes and each class decomposes in a specific unique way into the
irreducible representations of $\mathfrak{GU}_\Lambda$. Hence all
Arnold-Beltrami vector fields are in  correspondence with the irreps
of $\mathfrak{GU}_\Lambda$. Knowing the branching rules of such
irreps with respect to his various subgroups $\mathrm{H_i} \subset
\mathfrak{GU}_\Lambda$ and selecting the identity representation one
obtains Arnold-Beltrami vector fields invariant with respect to
those $\mathrm{H_i}$ for which we are able to find an identity irrep
$D_1$ in the branching rules. In this way we can classify all Arnold
Beltrami flows and also uncover their \textit{hidden symmetries}.
\par
Such a conclusion was already reached in \cite{Fre:2015mla}.
\par
As we recalled above, the authors of \cite{Fre:2015mla}  considered
in an extensive way the case of the cubic lattice and constructed
the corresponding Universal Classifying Group $\mathfrak{GU}_{cubic}
\, = \, \mathrm{G_{1536}}$. This latter is a finite group  of order
$|\mathrm{\mathrm{G_{1536}}}| \, = \, 1536$ which was studied in
full detail in \cite{Fre:2015mla}. All of its 37 irreducible
representations were derived and the associated character table was
also constructed. A large class of its subgroups $\mathrm{H_i
}\subset \mathrm{\mathrm{G_{1536}}}$ were also singled out and each
of them was studied systematically, by constructing their irreps and
character tables. This allowed the derivation of all the
\textit{branching rules} of the 37 irreps of
$\mathrm{\mathrm{G_{1536}}}$ with respect to the considered
subgroups which were displayed in dedicated tables in the appendices
of \cite{Fre:2015mla}. In the present paper one of the goals is that
of providing the same group theoretical lore for the case of the
hexagonal lattice which in \cite{Fre:2015mla} was only briefly
touched upon and sketched.
\par
Since the crystallographic lattices are more than two one might
think that covering these two cases is only part of the work.  It is
not so. Mastering the Universal Classifying Groups for the cubic and
hexagonal lattices is sufficient to provide the entire picture.
Indeed the crystallographic lattices in D=3 subdivide in just two
classes \cite{Aroyo,Souvignier,bravaislat}:
\begin{description}
  \item[A)] The lattices whose basis vectors $\mathbf{w}_\lambda$ provide an orthogonal
  basis (although not necessarily orthonormal):
  \begin{equation}\label{ortobambo}
    \left(\mathbf{w}_\lambda \, , \, \mathbf{w}_\mu\right) \, = \,
    a_\lambda^2 \, \delta_{\lambda\mu}
  \end{equation}
where $\delta_{\lambda\mu}$ is the Kronecker delta and $a_\lambda$
is the lattice spacing in direction $\lambda\, = \, 1,2,3$.
  \item[B)] The lattices whose basis vectors $\mathbf{w}_\lambda$
  are arranged as follows:
  \begin{eqnarray}\label{hexabambo}
    \left(\mathbf{w}_1 \, , \, \mathbf{w}_1\right)\, = \,\left(\mathbf{w}_2\, , \,
    \mathbf{w}_2\right) & = & a^2 \nonumber\\
    \left(\mathbf{w}_3 \, , \,
    \mathbf{w}_3\right) & = & b^2\nonumber\\
\left(\mathbf{w}_1 \, , \, \mathbf{w}_2\right)& = & a^2 \, \cos
\left[\frac{2\pi}{3} \right] \nonumber\\
\left(\mathbf{w}_1 \, , \, \mathbf{w}_3\right)\, =
\,\left(\mathbf{w}_2 \, , \,
    \mathbf{w}_3\right) & = & 0
  \end{eqnarray}
  $a$  being the lattice spacing in each horizontal plane spanned by
  $\mathbf{w}_{1,2}$
  which is endowed with a hexagonal tesselation and $b$ the lattice
  spacing in the third vertical direction.
\end{description}
The point groups pertaining to the lattices of class A) are:
\begin{equation}\label{Aclasspointgroups}
    \mathfrak{P}_{\Lambda_A} \, = \, \left ( \mathrm{C_2, \, C_4, \, Dih_2,
    \, Dih_4, \, T_{12}, \, O_{24}} \right )
\end{equation}
where $\mathrm{C_n}$ denotes the cyclic group of order $\mathrm{n}$,
$\mathrm{Dih_m}$ denotes the dihedral group of order $\mathrm{m}$
and $\mathrm{T_{12}}$ is the tetrahedral group, while
$\mathrm{O_{24}}$ is the already mentioned octahedral group. All the
point groups in the list (\ref{Aclasspointgroups}) are subgroups of
the maximal one $\mathrm{O_{24}}$.
\par
The point groups pertaining to the lattices of class B) are:
\begin{equation}\label{Bclasspointgroups}
    \mathfrak{P}_{\Lambda_B} \, = \, \left ( \mathrm{C_3, \, C_6, \, Dih_3,
    \, Dih_6} \right )
\end{equation}
All the point groups in the list (\ref{Bclasspointgroups}) are
subgroups of the maximal one $\mathrm{Dih_6}$.
\par
This fact has the important consequence that the Universal
Classifying Group for the cubic lattice contains as subgroups the
Universal Classifying Groups for all the other lattices of class A),
while the Universal Classifying group for the hexagonal lattice
contains as subgroups all the Universal Classifying Groups for the
lattices of class B). Since, as we explain below, the construction
of Beltrami fields is organized into irreducible representations of such classifying
groups, once we have the algorithm for the largest group we have
also that for all its subgroups.
\par
In the case of the cubic lattice, the main result of
\cite{Fre:2015mla} was the proof that the $\mathrm{O_{24}}$ orbits
in the cubic lattice arrange into $48$ equivalence classes, the
parameters of the corresponding Beltrami vector fields  filling all
the 37 irreducible representations of $\mathrm{\mathrm{G_{1536}}}$.
\subsection{The classical ABC flows} \label{abcperbambini}
 The following vector field:
\begin{equation}\label{bagcigaluppi}
  \mathbf{u}(x,y,z) \, = \, \mathcal{V}^{(ABC)}(x,y,z) \, \equiv \, \left(
\begin{array}{l}
 C \cos (2 \pi  y)+A \sin (2 \pi  z) \\
 A \cos (2 \pi  z)+B \sin (2 \pi  x) \\
 B \cos (2 \pi  x)+C \sin (2 \pi  y)
\end{array}
\right)
\end{equation}
which satisfies the Beltrami condition with eigenvalue $\lambda \, =
\, 1$ and which contains three real parameters $A,B,C$ defines  what
is known in the literature by the name of an ABC-flow
(Arnold-Beltrami-Childress) and during the last half century it was
the target of fantastically numerous investigations.
\par
Main motivation of the paper \cite{Fre:2015mla}  was to understand
the principles underlying the construction  of the ABC-flows in
order to use systematically such principles to construct and
classify all other Arnold-like Beltrami flows, deriving also, as a
bonus, their hidden discrete symmetries. For instance symmetries of
Beltrami flows have proved to be crucial in connection with their
use in modeling \textit{magneto-hydrodynamic fast
dynamos}\cite{zeldus},\cite{Dynamo},\cite{FFMF},\cite{Gilbert}. By
this words it is understood the mechanism that in a steady flow of
charged particles generates a large scale magnetic field whose
magnitude might be exponentially increasing with time. No analytic
results do exist on fast dynamos and all studies have been so far
numerical, yet while dealing with these latter, crucial
simplifications occur and optimization algorithms become available
if the steady flow possesses a large enough group $\mathcal{G}$ of
symmetries. In this case the magnetic field can be developed into
irreducible representations of  $\mathcal{G}$ and this facilitates
the numerical determination of growing rates of different modes. It
is important to stress that the linearized dynamo equations for the
magnetic field $\mathbf{B}$ coincide with the linearized equations
for perturbations around a steady flow. Therefore the same
development of perturbations into irreps of $\mathcal{G}$ is of
great relevance also for the study of fluid instabilities.
\par
As already stressed a much shorter sketch of the Hexagonal Lattice
was provided in \cite{Fre:2015mla} in order to emphasize the
generality of the applied methods, yet the authors did not address
the construction of the Universal Classifying Group  which is one of
the tasks addressed in the present paper.

\section{The spectrum of
the $\star \mathbf{d}$ operator on $\mathrm{T}^3$}
\label{fantasmabeltrami} The main issue of paper \cite{Fre:2015mla}
was the construction of vector fields defined over the three-torus
$T^3$ that are eigenstates of the $\star_g \mathrm{d}$ operator,
namely  solutions of the following equation:
\begin{eqnarray}
\star_g \mathrm{d}\Omega^{(n;I)} &=& \, m_{(n)} \, \Omega^{(n;I)} \nonumber\\
\Omega^{(n;I)}\left[V_{(m;J)} \right] &=& \delta^n_m \, \delta^I_J
\label{formaduale}
\end{eqnarray}
where $\mathrm{d}$ is the exterior differential, and $\star_g$ is
the Hodge-duality operator which, differently from the exterior
differential, can be defined only with reference to a given metric
$g$. By $\Omega^{(n;I)}$ we denote a one-form:
\begin{equation}\label{omegas}
    \Omega^{(n;I)} \, = \, \Omega^{(n;I)}_\mu \,dx^\mu
\end{equation}
which is declared to be dual to the vector field we are interested
in:
\begin{eqnarray}\label{vettocampo}
V_{(m;J)} & = & V_{(m;J)}^\mu \, \partial_\mu \nonumber\\
\Omega^{(n;I)}\left[V_{(m;J)} \right] & \equiv &  \Omega^{(n;I)}_\mu
\, V_{(m;J)}^\mu \, = \, \delta^n_m \, \delta^I_J
\end{eqnarray}
and by means of the composite index $(n;I)$ we make reference to the
quantized eigenvalues $m_{(n)}$ of the $\star_g \mathrm{d}$ operator
(ordered in increasing magnitude $|m_{(n)}|$) and to a basis of the
corresponding eigenspaces
\begin{equation}\label{superpongomega}
\star_g \mathrm{d}\Omega^{(n)} \, = \, \, m_{(n)} \, \Omega^{(n)}
\quad \Rightarrow \quad \Omega^{(n)}\, = \, \sum_{I=1}^{d_n} \,c_I
\, \Omega^{(n;I)}
\end{equation}
the symbol $d_n$ denoting the degeneracy of $|m_{(n)}|$ and $c_I$
being constant coefficients.
\par
Indeed, since $T^3$ is a compact manifold, the eigenvalues $m_{(n)}$
form a discrete set. Their values and their degeneracies are a
property of the metric $g$ introduced on it. Here we outline the
general procedure to construct the eigenfunctions of $\star_g
\mathrm{d}$, to calculate the eigenvalues and to determine  their
degeneracies. What follows is an elementary and straightforward
exercise in harmonic analysis.
\par
In tensor notation, equation (\ref{formaduale}) has the following
appearance:
\begin{equation}\label{tensoBeltra}
\frac{1}{2\,e} \, g_{\mu\nu} \, \epsilon^{\nu\rho\sigma} \partial_\rho
\Omega_\sigma \, = \, m \, \Omega_\mu\,,
\end{equation}
where $e\equiv \sqrt{{\rm det}(g_{\mu\nu})}$.
The equation written above was named Beltrami equation since it was
already considered by the great italian mathematician Eugenio
Beltrami in 1881 \cite{beltramus}, who presented one of its periodic
solutions previously constructed by  Gromeka in 1881. Such a
solution was inherited by Arnold and it is essentially the basis of
his Hydrodynamical Model. We will see that Arnold Model just
corresponds to the lowest eigenfunction of the $\star_g \,
\mathrm{d}$-operator in the case of the cubic lattice. Many more
similar models can be constructed choosing higher eigenvalues,
choosing irreducible representation of the point group  in their
eigenspaces or changing the lattice.
\par
Introducing the basis vectors of the dual lattice $\Lambda^\star$ we
can write:
\begin{equation}\label{bardacco1}
    \Omega \, = \, \Omega_\mu \, dr^\mu  \, = \, \Omega_\mu \, e^\mu_i \, dx^i \, = \, \Omega_i \, dx^i
\end{equation}
where $e^\mu_i$ are the components of the vectors $
{\mathbf{e}}^\mu$ in a standard orthogonal basis of $\mathbb{R}^3$
and
\begin{equation}\label{bardacco2}
    x^i \, = \, w_\mu^i \, r^\mu
\end{equation}
are a new set of euclidian coordinates obtained from the original
ones $r^\mu$ by means  of the components $w_\mu^i$ of the basis
vectors $ {\mathbf{w}}_\mu$ of the space lattice $\Lambda$.
Recalling that:
\begin{equation}\label{derivosugiu}
\partial_\mu \, =\, \frac{\partial}{\partial r^\mu} \, = \, w^\mu_i \, \partial_i \, = \, w^\mu_i \, \frac{\partial}{\partial x^i}
\end{equation}
with a little bit of straightforward algebra we can rewrite
eq.(\ref{formaduale}) in the equivalent universal way:
\begin{equation}\label{tensoBeltra2}
\frac{1}{2} \,  \epsilon_{ijk} \partial_j \Omega_k \, = \, \mu  \,
\Omega_i \quad ; \quad \mu \, = \, m
\end{equation}
The next task is that of constructing an ansatz
for the vector harmonics $Y_i( {\mathbf{x}})$ that are
eigenfunctions of  $\star_g \mathrm{d}$. Since such eigenfunctions
have to be well defined on $\mathrm{T}^3$, their general form is
necessarily the following one:
\begin{eqnarray}\label{harmogen}
Y_i\left( {\mathbf{k}}\, | \, {\mathbf{x}}\right) & = & v_i\left(
{\mathbf{k}}\right) \,\cos\left( 2\,\pi \,  {\mathbf{k}}\cdot
 {\mathbf{x}}\right)
\, +\, \omega_i\left( {\mathbf{k}}\right) \,\sin\left( 2\,\pi \,  {\mathbf{k}}\cdot  {\mathbf{x}}\right) \nonumber\\
 {\mathbf{k}} & \in & \Lambda^\star
\end{eqnarray}
The condition that the momentum $ {\mathbf{k}}$ lies in the dual
lattice guarantees that $Y_i( {\mathbf{x}})$ is periodic with
respect to the space lattice $\Lambda$. Indeed, by means of the very
definition of dual lattice (\ref{reticoloLastar}) it follows that:
\begin{equation}\label{periodico}
\forall \,  {\mathbf{q}} \, \in \, \Lambda \, : \quad Y_i\left(
{\mathbf{x}}\, + \,  {\mathbf{q}}\right) \, = \, Y_i\left(
{\mathbf{x}}\right)
\end{equation}
Considering next eq. (\ref{tensoBeltra2}) we immediately see that it
implies the further condition $\partial^i \, Y_i \, = \, 0$.
Imposing such a condition on the general ansatz (\ref{harmogen}) we
obtain:
\begin{equation}\label{transversocondo}
    {\mathbf{k}}\, \cdot \,  {\mathbf{v}}\left( {\mathbf{k}}\right) \, = \, 0 \quad ; \quad  {\mathbf{k}}\,
   \cdot \,  {\mathbf{\omega}}\left( {\mathbf{k}}\right) \, = \, 0
\end{equation}
which reduces the 6 parameters contained in the general ansatz
(\ref{harmogen}) to 4. Imposing next the very equation
(\ref{tensoBeltra2}) we get the following two conditions:
\begin{eqnarray}
  \mu \, v_i \left( {\mathbf{k}}\right) &=& \pi \, \epsilon_{ij\ell} \, k_j \,
  \omega_\ell \left( {\mathbf{k}}\right) \label{curlo1} \\
  \mu \, \omega_i \left( {\mathbf{k}}\right) &=& -\pi \,
  \epsilon_{ij\ell} \, k_j \, v_\ell \left( {\mathbf{k}}\right) \label{curlo2}
\end{eqnarray}
The two equations are self consistent if and only if the following
condition is verified:
\begin{equation}\label{spectra1}
    \mu^2 \, = \, \pi^2 \, \langle {\mathbf{k}}\, , \,  {\mathbf{k}}\rangle
\end{equation}
This trivial elementary calculation completely determines the
spectrum of the operator $\star_g \, \mathrm{d}$ on $\mathrm{T}_g^3$
endowed with the metric fixed by the choice of a lattice $\Lambda$.
The possible eigenvalues are provided by:
\begin{equation}\label{autovalore}
    m_\mathbf{k} \, = \, \pm \, \pi \, \, \sqrt{\langle {\mathbf{k}}\, ,
    \,  {\mathbf{k}}\rangle}  \quad \quad  {\mathbf{k}} \, \in \, \Lambda^\star
\end{equation}

The degeneracy of each eigenvalue is geometrically provided by
counting the number of intersection points of the dual lattice
$\Lambda^\star$ with a sphere whose center is in the origin and
whose radius is:
\begin{equation}\label{radius}
    r \, = \,  \sqrt{\langle {\mathbf{k}}\, , \,  {\mathbf{k}}\rangle}
\end{equation}
For a generic lattice the number of solutions of equation
(\ref{radius}) namely the number of intersection points of the
lattice with the sphere is just two: $\pm  {\mathbf{k}}$, so that
the typical degeneracy of each eigenvalue is just $2$. If the
lattice $\Lambda$ is one of the Bravais lattices admitting a non
trivial point group  $\mathrm{G}$, then the number of solutions of
eq.(\ref{radius})  increases since all lattice vectors $
{\mathbf{k}}$ that sit  in one orbit of  $\mathrm{G}$ have the same
norm and therefore are located on the same spherical surface. The
degeneracy of the $\star_g \, \mathrm{d}$ eigenvalue is precisely
the order of the corresponding $\mathrm{G}$-orbit in the dual
lattice $\Lambda^\star$. It is appropriate to note that the spectrum
of the $\star_g \, \mathrm{d}$ operator on vectors is just the
square root of the spectrum of the Laplacian operator on the same
space. Indeed if we have a scalar function $\Phi( {\mathbf{x}})$ on
$\mathrm{T}_g^3$ it admits the expansion in harmonics of the form:
\begin{eqnarray}\label{harmogen2}
Y\left( {\mathbf{k}}\, | \,  {\mathbf{x}}\right) & = & a\left(
{\mathbf{k}}\right) \, \cos\left( 2\,\pi \,  {\mathbf{k}}\cdot
 {\mathbf{x}}\right) \, +\, b\left( {\mathbf{k}}\right)
\,\sin\left( 2\,\pi \,  {\mathbf{k}}\cdot  {\mathbf{x}}\right) \nonumber\\
 {\mathbf{k}} & \in & \Lambda^\star
\end{eqnarray}
and calculating the laplacian we obtain
\begin{equation}\label{laplaccio}
\Delta_g \, Y\left( {\mathbf{x}}\right) \, \equiv  \, \frac{1}{4} \,
g^{\mu\nu}\, \partial_\mu \, \partial_\nu \, Y\left(
{\mathbf{x}}\right) \, = \,  \pi^2 \, \langle {\mathbf{k}}\, , \,
 {\mathbf{k}}\rangle \, Y\left( {\mathbf{x}}\right)
\end{equation}
In particular all the three components of the vector harmonic
(\ref{harmogen}) satisfy the above equation with the above
eigenvalue.
\subsection{The algorithm to construct Arnold Beltrami
Flows} \label{algoritmo} What we described in the previous
subsection provides a well defined algorithm to construct a series
of Arnold Beltrami flows that can be summarized in a few clear-cut
steps and it is quite suitable for a systematic  computer aided
implementation.
\par
The steps are the following ones:
\begin{description}
  \item[a)] Choose a Bravais Lattice $\Lambda$ with a non trivial proper point group  $\mathfrak{P}_\Lambda$.
  \item[b)] Construct the character table and the irreducible representations of $\mathfrak{P}_\Lambda$.
  \item[c)] Analyze the structure of orbits of $\mathfrak{P}_\Lambda$ on the lattice $\Lambda$ and determine
  the number  of lattice points  contained in each spherical layer $\mathfrak{S}_n$ of the dual lattice $\Lambda^\star$
  of quantized radius $r_n$:
\begin{eqnarray}\label{spherlayer}
     && {\mathbf{k}}_{(n)} \, \in \,  \mathfrak{S}_n \quad \Leftrightarrow \quad  \langle  {\mathbf{k}}_{(n)} \, , \,
      {\mathbf{k}}_{(n)}\rangle \, = \, r_n^2\nonumber\\
     && 2\,P_n  \, \equiv \, \left|  \mathfrak{S}_n \right|
\end{eqnarray}

The number of lattice points in each spherical layer is always even
since if $ {\mathbf{k}} \, \in \,
      \Lambda^\star$ also $- {\mathbf{k}} \, \in \, \Lambda^\star$ and obviously any vector and its negative have the same norm.
The spherical layer $\mathfrak{S}_n$ can be composed of one or of
more $\mathfrak{P}_\Lambda$-orbits. In any case it corresponds to a
fixed eigenvalue $m_n \, = \, \pi \, r_n$ of the $\star \,
\mathrm{d}$-operator.
  \item[d)] Construct the most general solution of the Beltrami equation with eigenvalue $m_n$ by using the individual harmonics
  constructed in eq. (\ref{harmogen}):
\begin{equation}\label{beltramusgen}
V_i\left( {\mathbf{x}}\right) \, = \, \sum_{ {\mathbf{x}}\, \in
\,\mathfrak{S}_n } \,Y_i\left( {\mathbf{k}}\, | \,
{\mathbf{x}}\right)
\end{equation}
Hidden in each harmonic $Y_i\left( {\mathbf{k}}\, | \,
{\mathbf{x}}\right)$ there are two parameters that are the remainder
of the six parameters $v_i\left( {\mathbf{k}}\right)$ and
$\omega_i\left( {\mathbf{k}}\right)$ after conditions
(\ref{transversocondo},\ref{curlo1},\ref{curlo2}) have been imposed.
This would amount to a total of $4\, P_n$ parameters, yet, since the
trigonometric functions $\cos(\theta)$ and $\sin(\theta)$ are mapped
into plus or minus themselves under change of sign of their argument
and since each spherical layer $\mathfrak{S}_n $ contains lattice
vectors in pairs $\pm  {\mathbf{k}}$ , it follows that the number of
independent parameters is always reduced to $2 P_n$. Hence, at the
end of the construction encoded in eq. (\ref{beltramusgen}), we have
a Beltrami vector depending on a set of $2 P_n$ parameters that we
can call $F_I$ and consider as the components of $2 \,
P_n$-component vector $\mathbf{F}$. Ultimately we have an object of
the following form:
\begin{equation}\label{gomez}
    \mathbf{V}\left( {\mathbf{x}}\, | \, \mathbf{F}\right)
\end{equation}
which under the point group   $\mathfrak{P}_\Lambda$ necessarily
transforms in the following way:
\begin{equation}\label{Rtrasformogen}
    \forall \, \gamma \, \in \, \mathfrak{P}_\Lambda \, : \quad \gamma^{-1} \,\cdot \,
    \mathbf{V}\left(\gamma \,\cdot \, {\mathbf{x}}\, | \, \mathbf{F}\right)
    \, = \, \mathbf{V}\left( {\mathbf{x}}\, | \, \mathfrak{R}[\gamma] \,\cdot \, \mathbf{F}\right)
\end{equation}
where $\mathfrak{R}[\gamma]$ are $2\, P_n \times 2\, P_n$ matrices
that form a representation of $\mathfrak{P}_\Lambda$.
Eq.(\ref{Rtrasformogen}) is necessarily true because any rotation
$\gamma \, \in \, \mathfrak{P}_\Lambda$ permutes the elements of
$\mathfrak{S}_n$ among themselves.
\item[e)] Decompose the representation $\mathfrak{R}[\gamma]$ into irreducible representations of $\mathfrak{P}_\Lambda$.
Each irreducible subspace $\mathbf{f}_{p}$ of the $2 P_n$ parameter
space $\mathbf{F}$ defines an Arnold--Beltrami Flow:
\begin{equation}\label{arnoldosim}
\frac{\mathrm{d}}{\mathrm{d}t} \,  {\mathbf{x}}(t) \, = \,
\mathbf{V}\left( {\mathbf{x}}(t)\, | \, \mathbf{f}_p\right)
\end{equation}
which is worth to analyze.
\end{description}
An obvious question which arises in connection with such a
constructive algorithm is the following: how many Arnold--Beltrami
flows are there? At first sight it seems that there is an infinite
number of such systems since we can arbitrarily increase the radius
of the spherical layer and on each new layer it seems that we have
new models. Let us however observe that if on two different
spherical layers $\mathfrak{S}_{n_1}$ and $\mathfrak{S}_{n_2}$ there
are two orbits of lattice vectors $\mathcal{O}_1$ and
$\mathcal{O}_2$ that have the same order
\begin{equation}\label{parorder}
   \ell \, = \,  \left| \mathcal{O}_1\right | \, = \, \left| \mathcal{O}_2\right |
\end{equation}
and furthermore all vectors ${\mathbf{k}}_{(n_2)} \, \in
\,\mathcal{O}_2$ are simply proportional to their analogues in orbit
$\mathcal{O}_1$:
\begin{equation}\label{proporzio}
{\mathbf{k}}_{(n_2)} \, = \, \lambda \, {\mathbf{k}}_{(n_1)} \quad ;
\quad \lambda \, \in \, \mathbb{Z}
\end{equation}
then we can conclude that:
\begin{equation}\label{riduzione}
\mathbf{V}_{(n_2)}\left( {\mathbf{x}}\, | \, \mathbf{f}_p\right) \,
= \,  \mathbf{V}_{(n_1)}\left(\, \lambda \, {\mathbf{x}}\, | \,
\mathbf{f}_p\right)
\end{equation}
By redefining  the coordinate fields $\lambda \, {\mathbf{x}} \, =
\,  {\mathbf{x}}^\prime$ and rescaling time $t$ the two differential
systems (\ref{arnoldosim}) respectively  constructed from  layer
$n_1$ and layer $n_2$ can be identified.
\par
As it was demonstrated  in \cite{Fre:2015mla} analyzing the case of
the cubic lattice and the orbits of the octahedral  group there is
always a finite number of $\mathfrak{P}_\Lambda$-orbit type on each
lattice $\Lambda$. There is a  maximal orbit $\mathcal{O}_{max}$
that has order equal to the order of the point group :
\begin{equation}\label{orbitmax}
\left| \mathcal{O}_{max}\right| \, = \, \left|
\mathfrak{P}_\Lambda\right|
\end{equation}
and there are a few shortened orbits $\mathcal{O}_{i}$ ($i=\,1,\dots
,\, s$) that have a smaller order:
\begin{equation}\label{carisma}
\ell_i \, = \, \left| \mathcal{O}_{i}\right| \, < \, \left|
\mathfrak{P}_\Lambda\right|
\end{equation}
The fascinating property is that for the shortened orbits which seem
to play an analogous role in this context to that of BPS states in
another context, property (\ref{proporzio}) is always true. The
vectors pertaining to the same orbit $\mathcal{O}_i$ in different
spherical layers are always the same up to a multiplicative factor.
Hence from the shortened orbits it was shown in \cite{Fre:2015mla}
that one always obtains a finite number of Arnold--Beltrami flows.
It remained the case of the maximal orbit for which property
(\ref{proporzio}) is not necessarily imposed. How many independent
flows do we obtain considering all the layers? The answer to the
posed question is hidden in number theory. Indeed one has to analyze
how many different type of triplets of integer numbers satisfy
Diophantine equations of the Fermat type. In section
\ref{triplettoni} we review the answer obtained in
\cite{Fre:2015mla} providing a systematic classification of such
triplets for the cubic lattice.
\par
Actually that classification is a classification of sublattices of
the cubic lattice and each sublattice is associated with irreducible
representations of the Universal Classifying Group
$\mathfrak{GU}_\Lambda$.
\par
Such result demonstrated that there is a finite number of
Arnold--Beltrami flows and each of them can be promoted to a
definite type of exact solutions of the Navier Stokes equations
depending on a finite number of parameters that acquire a dependence
on the momenta and are, in this way, identified with Fourier
coefficients in a Fourier series expansion of the initial
conditions.
\chapter{Group Theory Foundations} \label{gruppafunda}
In order to make the present paper self-consistent and better
highlight the interpretation of several of the results obtained in
\cite{Fre:2015mla}, that here are clarified in a more systematic way
and extended from the cubic to the hexagonal case, we review the
main group theoretical ingredients utilized in \cite{Fre:2015mla} to
derive the \textit{Universal Classifying Group},  whose very notion
in the present paper is made more precise in view of \textit{exact
sequences} and \textit{finite group cohomology}.
\par
Skipping generalities we just remind the reader of what was already
presented in eq.s(\ref{Aclasspointgroups},\ref{Bclasspointgroups}),
namely that in three dimensions the available \textit{Lattice Point
Groups} $\mathfrak{P}_\Lambda$ are either the cyclic groups
$\mathrm{C}_h \sim\mathbb{Z}_h$ with $h=2,3,4,6$ or the dihedral
groups $\mathrm{Dih}_h$ with $h=2,3,4,6$ or the tetrahedral group
$\mathrm{T_{12}}\sim \mathrm{A_4}$ or the octahedral group
$\mathrm{O_{24}}\sim \mathrm{S_4}$.
\section{The cubic lattice and the octahedral point group $\mathrm{O_{24}}$}
\label{reticolocubico} The case of the cubic lattice was analyzed in
depth in \cite{Fre:2015mla}. We review and repeat here a good deal
of the results of that paper for three reasons:
\begin{enumerate}
  \item We need to revise the conventions and the notations in order
  to make clear how the upgrading to the complete Navier Stokes
  equations is achieved in practice.
  \item Since a large part of the results to be obtained, classified
  and visualized necessarily depends on the use of MATHEMATICA codes
  that derive from those developed in 2014-2015 by means of a
  systematic reorganization of the routines and subroutines and by
  a transcription from MATHEMATICA 5.2 to MATHEMATICA 12, it is of
  vital importance to utilize a well defined and already established
  set of conventions and nomenclature.
  \item The cubic lattice case constitutes the paradigm for the
  development of the same lore in the case of the hexagonal lattice which is
  a  goal of the present paper.
\end{enumerate}
Hence, within the general frame presented above, let us review the
cubic lattice case.
\par
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=70mm]{CubLatPSGen.jpg}
\end{center}
\caption{\it  A view of the self-dual cubic lattice}
\label{cubicogenerale}
\end{figure}
The self-dual cubic lattice (momentum and space lattice at the same
time) is displayed in fig.\ref{cubicogenerale}.
\par
The basis vectors of the cubic lattice $\Lambda_{cubic}$ are :
\begin{equation}\label{cubicobase}
{\mathbf{w}}_1 \, = \, \{1,0,0\} \quad ; \quad {\mathbf{w}}_2 \, =
\, \{0,1,0\} \quad ; \quad {\mathbf{w}}_3 \, = \, \{0,0,1\}
\end{equation}
which implies that the metric is just the Kronecker delta:
\begin{equation}\label{Kronecca}
    g_{\mu\nu} \, = \, \delta_{\mu\nu}
\end{equation}
and the basis vectors ${\mathbf{e}}^\mu$ of the dual lattice
$\Lambda_{cubic}^\star$ coincide with those of the lattice
$\Lambda$. Hence the cubic lattice is self-dual:
\begin{equation}\label{autoduali}
     {\mathbf{w}}_\mu \, = \,  {\mathbf{e}}^\mu \quad \Rightarrow \quad \Lambda_{cubic} \, = \, \Lambda^\star_{cubic}
\end{equation}
The subgroup of the proper rotation group which maps the cubic
lattice into itself is the octahedral  group $\mathrm{O_{24}}$ whose
order is 24. In the next subsection we recall its structure.
\subsection{Structure of the  Octahedral Group $\mathrm{O_{24}}\sim \mathrm{S_4}$}
\label{ottostruttura} Abstractly the octahedral Group
$\mathrm{O_{24}}\sim \mathrm{S_{4}}$ is isomorphic to the symmetric
group of permutations of 4 objects. It is defined by the following
generators and relations:
\begin{equation}\label{octapresa}
T, \, S \quad : \quad    T^3 \, = \, \mbox{\bf e} \quad ; \quad S^2
\, = \, \mbox{\bf e} \quad ; \quad (S\,T)^4 \, = \, \mbox{\bf e}
\end{equation}
On the other hand $\mathrm{O_{24}}$ is a finite, discrete subgroup
of the three-dimensional rotation group and any $\gamma \, \in \,
\mathrm{O_{24}}\, \subset \, \mathrm{SO(3)}$ of its 24 elements can
be uniquely identified by its action on the coordinates $x,y,z$,  as
it is displayed below:
\begin{equation}\label{nomiOelemen}
\begin{array}{cc}
\begin{array}{|c|rcl|}
\hline
\mbox{\bf e} & 1_1 & = & \{x,y,z\} \\
 \hline
\null & 2_1 & = & \{-y,-z,x\} \\
\null &  2_2 & = & \{-y,z,-x\} \\
\null & 2_3 & = & \{-z,-x,y\} \\
C_3 & 2_4 & = & \{-z,x,-y\} \\
 \null &2_5 & = & \{z,-x,-y\} \\
\null & 2_6 & = & \{z,x,y\} \\
\null & 2_7 & = & \{y,-z,-x\} \\
\null & 2_8 & = & \{y,z,x\} \\
 \hline
\null & 3_1 & = & \{-x,-y,z\} \\
C_4^2 & 3_2 & = & \{-x,y,-z\} \\
\null & 3_3 & = & \{x,-y,-z\} \\
 \hline
\end{array} & \begin{array}{|c|rcl|}
\hline
\null & 4_1 & = & \{-x,-z,-y\} \\
\null & 4_2 & = & \{-x,z,y\} \\
C_2 &4_3 & = & \{-y,-x,-z\} \\
\null & 4_4 & = & \{-z,-y,-x\} \\
\null & 4_5 & = & \{z,-y,x\} \\
\null & 4_6 & = & \{y,x,-z\} \\
 \hline
\null & 5_1 & = & \{-y,x,z\} \\
\null & 5_2 & = & \{-z,y,x\} \\
C_4 & 5_3 & = & \{z,y,-x\} \\
\null & 5_4 & = & \{y,-x,z\} \\
\null & 5_5 & = & \{x,-z,y\} \\
\null & 5_6 & = & \{x,z,-y\}\\
 \hline
\end{array} \\
\end{array}
\end{equation}
As one sees from the above list the 24 elements are distributed into
5 conjugacy classes mentioned in the first column of the table,
according to a nomenclature which is standard in the chemical
literature on crystallography. The relation between the abstract and
concrete presentation of the octahedral  group is obtained by
identifying in the list (\ref{nomiOelemen}) the generators $T$ and
$S$ mentioned in eq. (\ref{octapresa}). Explicitly we have:
\begin{equation}\label{generatiTS}
    T \, = \, 2_8 \, = \, \left(
\begin{array}{lll}
 0 & 1 & 0 \\
 0 & 0 & 1 \\
 1 & 0 & 0
\end{array}
\right)\quad ; \quad S \, = \, 4_6 \, =\left(
\begin{array}{lll}
 0 & 1 & 0 \\
 1 & 0 & 0 \\
 0 & 0 & -1
\end{array}
\right)
\end{equation}
All other elements are reconstructed from the above two using the
multiplication table of the group which is displayed below: {\small
\begin{equation}\label{Omultipla}
\begin{array}{|l|llllllllllllllllllllllll|}
\hline
 \null &   1_1 & 2_1 & 2_2 & 2_3 & 2_4 & 2_5 & 2_6 & 2_7 & 2_8 & 3_1 & 3_2 & 3_3 & 4_1 & 4_2 & 4_3 & 4_4 & 4_5 & 4_6 & 5_1 & 5_2 & 5_3 & 5_4 & 5_5 & 5_6 \\
 \hline
 1_1 &   1_1 & 2_1 & 2_2 & 2_3 & 2_4 & 2_5 & 2_6 & 2_7 & 2_8 & 3_1 & 3_2 & 3_3 & 4_1 & 4_2 & 4_3 & 4_4 & 4_5 & 4_6 & 5_1 & 5_2 & 5_3 & 5_4 & 5_5 & 5_6 \\
 2_1 &   2_1 & 2_5 & 2_4 & 3_3 & 3_2 & 1_1 & 3_1 & 2_6 & 2_3 & 2_7 & 2_2 & 2_8 & 5_3 & 4_4 & 5_6 & 4_6 & 5_4 & 4_2 & 4_1 & 4_3 & 5_1 & 5_5 & 4_5 & 5_2 \\
 2_2 &   2_2 & 2_6 & 2_3 & 1_1 & 3_1 & 3_3 & 3_2 & 2_5 & 2_4 & 2_8 & 2_1 & 2_7 & 4_5 & 5_2 & 5_5 & 5_4 & 4_6 & 4_1 & 4_2 & 5_1 & 4_3 & 5_6 & 5_3 & 4_4 \\
 2_3 &   2_3 & 3_2 & 1_1 & 2_2 & 2_8 & 2_7 & 2_1 & 3_3 & 3_1 & 2_4 & 2_6 & 2_5 & 4_6 & 5_1 & 5_3 & 5_6 & 4_1 & 4_5 & 5_2 & 4_2 & 5_5 & 4_4 & 4_3 & 5_4 \\
 2_4 &   2_4 & 3_1 & 3_3 & 2_1 & 2_7 & 2_8 & 2_2 & 1_1 & 3_2 & 2_3 & 2_5 & 2_6 & 5_4 & 4_3 & 4_5 & 5_5 & 4_2 & 5_3 & 4_4 & 4_1 & 5_6 & 5_2 & 5_1 & 4_6 \\
 2_5 &   2_5 & 1_1 & 3_2 & 2_8 & 2_2 & 2_1 & 2_7 & 3_1 & 3_3 & 2_6 & 2_4 & 2_3 & 5_1 & 4_6 & 5_2 & 4_2 & 5_5 & 4_4 & 5_3 & 5_6 & 4_1 & 4_5 & 5_4 & 4_3 \\
 2_6 &   2_6 & 3_3 & 3_1 & 2_7 & 2_1 & 2_2 & 2_8 & 3_2 & 1_1 & 2_5 & 2_3 & 2_4 & 4_3 & 5_4 & 4_4 & 4_1 & 5_6 & 5_2 & 4_5 & 5_5 & 4_2 & 5_3 & 4_6 & 5_1 \\
 2_7 &   2_7 & 2_3 & 2_6 & 3_1 & 1_1 & 3_2 & 3_3 & 2_4 & 2_5 & 2_1 & 2_8 & 2_2 & 5_2 & 4_5 & 4_2 & 5_1 & 4_3 & 5_6 & 5_5 & 5_4 & 4_6 & 4_1 & 4_4 & 5_3 \\
 2_8 &   2_8 & 2_4 & 2_5 & 3_2 & 3_3 & 3_1 & 1_1 & 2_3 & 2_6 & 2_2 & 2_7 & 2_1 & 4_4 & 5_3 & 4_1 & 4_3 & 5_1 & 5_5 & 5_6 & 4_6 & 5_4 & 4_2 & 5_2 & 4_5 \\
 3_1 &   3_1 & 2_8 & 2_7 & 2_6 & 2_5 & 2_4 & 2_3 & 2_2 & 2_1 & 1_1 & 3_3 & 3_2 & 5_6 & 5_5 & 4_6 & 5_3 & 5_2 & 4_3 & 5_4 & 4_5 & 4_4 & 5_1 & 4_2 & 4_1 \\
 3_2 &   3_2 & 2_7 & 2_8 & 2_5 & 2_6 & 2_3 & 2_4 & 2_1 & 2_2 & 3_3 & 1_1 & 3_1 & 5_5 & 5_6 & 5_4 & 4_5 & 4_4 & 5_1 & 4_6 & 5_3 & 5_2 & 4_3 & 4_1 & 4_2 \\
 3_3 &   3_3 & 2_2 & 2_1 & 2_4 & 2_3 & 2_6 & 2_5 & 2_8 & 2_7 & 3_2 & 3_1 & 1_1 & 4_2 & 4_1 & 5_1 & 5_2 & 5_3 & 5_4 & 4_3 & 4_4 & 4_5 & 4_6 & 5_6 & 5_5 \\
 4_1 &   4_1 & 5_4 & 4_6 & 4_5 & 5_3 & 5_2 & 4_4 & 5_1 & 4_3 & 5_5 & 5_6 & 4_2 & 1_1 & 3_3 & 2_8 & 2_6 & 2_3 & 2_2 & 2_7 & 2_5 & 2_4 & 2_1 & 3_1 & 3_2 \\
 4_2 &   4_2 & 4_6 & 5_4 & 5_3 & 4_5 & 4_4 & 5_2 & 4_3 & 5_1 & 5_6 & 5_5 & 4_1 & 3_3 & 1_1 & 2_7 & 2_5 & 2_4 & 2_1 & 2_8 & 2_6 & 2_3 & 2_2 & 3_2 & 3_1 \\
 4_3 &   4_3 & 5_3 & 5_2 & 5_6 & 4_2 & 5_5 & 4_1 & 4_5 & 4_4 & 4_6 & 5_1 & 5_4 & 2_6 & 2_4 & 1_1 & 2_8 & 2_7 & 3_1 & 3_2 & 2_2 & 2_1 & 3_3 & 2_5 & 2_3 \\
 4_4 &   4_4 & 4_2 & 5_5 & 5_1 & 5_4 & 4_6 & 4_3 & 5_6 & 4_1 & 5_2 & 4_5 & 5_3 & 2_8 & 2_1 & 2_6 & 1_1 & 3_2 & 2_5 & 2_3 & 3_1 & 3_3 & 2_4 & 2_2 & 2_7 \\
 4_5 &   4_5 & 5_6 & 4_1 & 4_6 & 4_3 & 5_1 & 5_4 & 4_2 & 5_5 & 5_3 & 4_4 & 5_2 & 2_2 & 2_7 & 2_4 & 3_2 & 1_1 & 2_3 & 2_5 & 3_3 & 3_1 & 2_6 & 2_8 & 2_1 \\
 4_6 &   4_6 & 4_4 & 4_5 & 4_1 & 5_5 & 4_2 & 5_6 & 5_2 & 5_3 & 4_3 & 5_4 & 5_1 & 2_3 & 2_5 & 3_1 & 2_1 & 2_2 & 1_1 & 3_3 & 2_7 & 2_8 & 3_2 & 2_4 & 2_6 \\
 5_1 &   5_1 & 4_5 & 4_4 & 5_5 & 4_1 & 5_6 & 4_2 & 5_3 & 5_2 & 5_4 & 4_3 & 4_6 & 2_5 & 2_3 & 3_3 & 2_7 & 2_8 & 3_2 & 3_1 & 2_1 & 2_2 & 1_1 & 2_6 & 2_4 \\
 5_2 &   5_2 & 4_1 & 5_6 & 4_3 & 4_6 & 5_4 & 5_1 & 5_5 & 4_2 & 4_4 & 5_3 & 4_5 & 2_7 & 2_2 & 2_5 & 3_3 & 3_1 & 2_6 & 2_4 & 3_2 & 1_1 & 2_3 & 2_1 & 2_8 \\
 5_3 &   5_3 & 5_5 & 4_2 & 5_4 & 5_1 & 4_3 & 4_6 & 4_1 & 5_6 & 4_5 & 5_2 & 4_4 & 2_1 & 2_8 & 2_3 & 3_1 & 3_3 & 2_4 & 2_6 & 1_1 & 3_2 & 2_5 & 2_7 & 2_2 \\
 5_4 &   5_4 & 5_2 & 5_3 & 4_2 & 5_6 & 4_1 & 5_5 & 4_4 & 4_5 & 5_1 & 4_6 & 4_3 & 2_4 & 2_6 & 3_2 & 2_2 & 2_1 & 3_3 & 1_1 & 2_8 & 2_7 & 3_1 & 2_3 & 2_5 \\
 5_5 &   5_5 & 4_3 & 5_1 & 4_4 & 5_2 & 5_3 & 4_5 & 4_6 & 5_4 & 4_1 & 4_2 & 5_6 & 3_2 & 3_1 & 2_2 & 2_4 & 2_5 & 2_8 & 2_1 & 2_3 & 2_6 & 2_7 & 3_3 & 1_1 \\
 5_6 &   5_6 & 5_1 & 4_3 & 5_2 & 4_4 & 4_5 & 5_3 & 5_4 & 4_6 & 4_2 & 4_1 & 5_5 & 3_1 & 3_2 & 2_1 & 2_3 & 2_6 & 2_7 & 2_2 & 2_4 & 2_5 & 2_8 & 1_1 & 3_3\\
 \hline
\end{array}
\end{equation}
} This observation is important in relation with representation
theory. Any linear representation of the group is uniquely specified
by giving the matrix representation of the two generators $T=2_8$
and $S=4_6$. In the sequel this will be extensively utilized in the
compact codification of the reducible representations that emerge in
our calculations.
\subsection{Irreducible representations of the Octahedral Group}
\label{ottoirreppi}
There are five conjugacy classes in $\mathrm{O}_{24}$ and therefore
according to theory there are five irreducible representations of
the same group, that we name $D_i$, $i\, =\, 1,\dots, 5$. Let us
briefly describe them.
\subsubsection{$D_1$ : the identity representation}
The identity representation which exists for all groups is that one
where to each element of $\mathrm{O}$ we associate the number $1$
\begin{equation}\label{identD1}
    \forall \, \gamma \, \in \, \mathrm{O}_{24} \,\, : \quad D_1(\gamma) \, = \, 1
\end{equation}
Obviously the character of such a representation is:
\begin{equation}\label{caretterusOD1}
    \chi_1 \, = \, \{1,1,1,1,1\}
\end{equation}
\subsubsection{$D_2$ : the quadratic Vandermonde representation}
The representation $D_2$ is also one-dimensional. It is constructed
as follows. Consider the following polynomial of order six in the
coordinates of a point in $\mathbb{R}^3$ or $\mathrm{T}^3$:
\begin{equation}\label{vPol}
    \mathfrak{V}(x,y,z) \, = \, (x^2 - y^2) \, (x^2 - z^2) \, (y^2 - z^2)
\end{equation}
As one can explicitly check under the transformations of the
octahedral  group listed in eq.(\ref{nomiOelemen}) the polynomial
$\mathfrak{V}(x,y,z)$ is always mapped into itself modulo an overall
sign. Keeping track of such a sign provides the form of the second
one-dimensional representation whose character is explicitly
calculated to be the following one:
\begin{equation}\label{caretterusOD2}
    \chi_1 \, = \, \{1,1,1,-1,-1\}
\end{equation}
\subsubsection{$D_3$ : the two-dimensional representation}
The representation $D_3$ is two-dimensional and it corresponds to a
homomorphism:
\begin{equation}\label{D3map1}
    D_3 \, : \quad \mathrm{O}_{24} \, \rightarrow \, \mathrm{SL(2,\mathbb{Z})}
\end{equation}
which associates to each element of the octahedral group a $2 \times
2$ integer valued matrix of determinant one. The homomorphism is
completely specified by giving the two matrices representing the two
generators:
\begin{equation}\label{D3map2}
    D_3(T) \, = \,
\left(
\begin{array}{ll}
 0 & 1 \\
 -1 & -1
\end{array}
\right) \quad ; \quad D_3(S) \, = \, \left(
\begin{array}{ll}
 0 & 1 \\
 1 & 0
\end{array}
\right)
\end{equation}
The character vector of $D_2$ is easily calculated from the above
information and we have:
\begin{equation}\label{caretterusOD3}
    \chi_3 \, = \, \{2,-1,2,0,0\}
\end{equation}
\subsubsection{$D_4$ : the three-dimensional defining representation}
The three dimensional representation $D_4$ is simply the defining
representation, where the generators $T$ and $S$ are given by the
matrices in eq.(\ref{generatiTS}).
\begin{equation}\label{D4map}
    D_4(T)\, = \, T \quad ; \quad D_4(S) \, = \, S
\end{equation}
From this information the characters are immediately calculated and
we get:
\begin{equation}\label{caretterusOD4}
    \chi_3 \, = \, \{3,0,-1,-1,1\}
\end{equation}
\subsubsection{$D_5$ : the three-dimensional unoriented representation}
The three dimensional representation $D_5$ is simply that one  where
the generators $T$ and $S$ are given by the following  matrices:
\begin{equation}
D_5(T) \, =  \, \left(
\begin{array}{lll}
 0 & 1 & 0 \\
 0 & 0 & 1 \\
 1 & 0 & 0
\end{array}
\right)\quad ; \quad D_5(S) \, = \, \left(
\begin{array}{lll}
 0 & 1 & 0 \\
 1 & 0 & 0 \\
 0 & 0 & 1
\end{array}
\right)
\end{equation}
From this information the characters are immediately calculated and
we get:
\begin{equation}\label{caretterusOD4bis}
    \chi_5 \, = \, \{3,0,-1,1,-1\}
\end{equation}
\begin{table}[!hbt]
  \centering
  \begin{eqnarray*}
   \begin{array}{||l||ccccc||}
    \hline
    \hline
{\begin{array}{cc} \null &\mbox{Class}\\
\mbox{Irrep} & \null\\
\end{array}} & \{\mbox{\bf e},1\} & \left\{C_3,8\right\} & \left\{C_4^2,3\right\} & \left\{C_2,6\right\} & \left\{C_4,6\right\} \\
\hline
 \hline
 D_1 \, , \quad \chi_1 \, = \,& 1 & 1 & 1 & 1 & 1 \\
D_2 \, , \quad \chi_2 \, = \,& 1 & 1 & 1 & -1 & -1 \\
 D_3 \, , \quad \chi_3 \, = \, & 2 & -1 & 2 & 0 & 0 \\
 D_4 \, , \quad \chi_4 \, = \, & 3 & 0 & -1 & -1 & 1 \\
D_5 \, , \quad \chi_5 \, = \, & 3 & 0 & -1 & 1 & -1\\
 \hline
 \hline
\end{array}
\end{eqnarray*}
  \caption{Character Table of the proper Octahedral Group}\label{caratteriO}
\end{table}
The table of characters is summarized in eq.(\ref{caratteriO}).
\section{The hexagonal lattice and the dihedral group $\mathrm{Dih_6}$}
\label{hexareticolo} We come next to a  discussion of the hexagonal
lattice. Since in this section all considered representations are
relative to the point group  we simplify the notation mentioning the
irreps only as $D_1, \dots, D_6$ without writing in square brackets
the group.
\subsection{The hexagonal lattice}
The basis vectors of the hexagonal space  lattice $\Lambda_{Hex}$
are the following ones :
\begin{equation}\label{Hexagbase}
    {\mathbf{w}}_1 \, = \, \left\{\sqrt{2},0,0\right\} \quad ; \quad {\mathbf{w}}_2 \, = \,
    \left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},0\right\} \quad ; \quad {\mathbf{w}}_3 \, = \, \left\{0,0,\sqrt{2}\right\}
\end{equation}
which implies that the metric is the following non diagonal one:
\begin{equation}\label{nonkronecca}
    g_{\mu\nu} \, = \, \left(
\begin{array}{ccc}
 2 & -1 & 0 \\
 -1 & 2 & 0 \\
 0 & 0 & 2 \\
\end{array}
\right)
\end{equation}
The basis vectors $ {\mathbf{e}}^\mu$ of the dual momentum lattice
$\Lambda_{Hex}^\star$ do not coincide with those of the lattice
$\Lambda_{Hex}$. They are the following ones:
\begin{equation}\label{Hexagdualbas}
    {\mathbf{e}}^1 \, = \, \left\{\frac{1}{\sqrt{2}},\frac{1}{\sqrt{6}},0\right\} \quad ;
    \quad {\mathbf{e}}^2 \, = \,\left\{0,\sqrt{\frac{2}{3}},0\right\} \quad ;
    \quad {\mathbf{e}}^3 \, = \,\left\{0,0,\frac{1}{\sqrt{2}}\right\}
\end{equation}
so that the space lattice is now a proper subgroup of its dual
$\Lambda_{Hex}^\star$, named also the \textit{momentum-lattice}. In
order to understand the structure of the hexagonal lattice one ought
to consider first the hexagonal tesselation of a plane that is
generated by the first two basis vectors ${\mathbf{w}}_{1,2}$.
\par
To this effect it is convenient to look at fig.\ref{Hexagtessere}
\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=150mm]{robapiattaconpesci.png}
\caption{{\it  A view of the hexagonal tesselation of the plane. The
hexagonal two dimensional lattice coincides with the $A_2$ root
lattice. Indeed the projection on the plane of the two basis vectors
$\mathbf{w}_1$ and $\mathbf{w}_2$ (the two blue vectors) are the two
simple roots of the $A_2$ Lie algebra. Each point of the lattice can
be regarded as the center of a regular hexagon whose vertices are
the first nearest neighbors. These hexagons provide a tesselation of
the infinite plane. } \label{Hexagtessere}}
\end{center}
\end{figure}
The space lattice which provides a tiling of the plane by means of
regular hexagons coincides with the root lattice of the  $A_2$ Lie
algebra its generators being the two simple roots $\alpha_{1,2}$.
\par
The plane projection of the dual lattice $\Lambda^\star_{Hex}$ is
just the weight lattice of $A_2$ the plane projection of the basis
vectors $\mathbf{e}_{1,2}$ being just the fundamental weights
$\lambda_{1,2}$. This is illustrated in the next fig.\ref{pesorete}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=90mm]{comparhexagplanar.png}
\caption{{\it  Illustration of the dual momentum lattice of the
hexagonal lattice in the plane. The red circles are the points of
the momentum lattice, while the blue ones are the points of the
space lattice. In the finite portions of the two lattices that we
show in this picture the black points are the common ones. As we see
each point of the space--lattice is surrounded by two hexagons; the
vertices of the smaller hexagon are moment-lattice points that do
not belong to space-lattice, while the vertices of the bigger
hexagon are the space-lattice nearest neighbors, as already remarked
in the caption of fig.\ref{Hexagtessere}.} \label{pesorete}}
\end{center}
\end{figure}
There it is clearly shown that the space lattice is a sublattice of
the dual momentum lattice.
\par
The three-dimensional hexagonal lattice is obtained by adjoining an
infinite number of equally spaced planes each tiled in the way shown
in fig.s \ref{Hexagtessere} and \ref{pesorete}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=70mm]{reticoloSHexag.png}
\includegraphics[height=70mm]{retPhexag.png}
\caption{\it  A view of the hexagonal space lattice $\Lambda_{Hex}$
(blue points on the left) and momentum momentum lattice
$\Lambda_{Hex}^\star$ (red points on the right)} \label{HexagLatPS}
\end{center}
\end{figure}
A view of the resulting three dimensional lattices is provided in
fig.\ref{HexagLatPS}.
\subsection{The point group $\mathrm{Dih_6}$}
The subgroup of the proper rotation group which maps the cubic
lattice into itself is the dihedral group $\mathrm{Dih}_6$ whose
order is 12. In the next lines we recall its structure.
\par
Abstractly the dihedral $\mathrm{Dih_6}$ group is defined by the
following generators and relations:
\begin{equation}\label{presentaD6}
    A \; , \; B \quad : \quad A^6 \, = \, \mbox{\bf e} \quad ; \quad B^2 \, = \, \mbox{\bf e} \quad ;
    \quad \left( BA\right)^2 \, = \, \mbox{\bf e}
\end{equation}
Explicitly in three dimensions we can take the following
matrix-representation for the generators of
$\mathrm{\mathrm{Dih_6}}$:
\begin{equation}\label{d3repD6}
    A \, = \, \left(
\begin{array}{lll}
 \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \\
 -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\
 0 & 0 & 1
\end{array}
\right) \quad ; \quad B \, = \, \left(
\begin{array}{lll}
 -1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & -1
\end{array}
\right)
\end{equation}
The group generated by the above generators has 12 elements that can
be arranged into 6 conjugacy classes, as it it is displayed in table
\ref{nomiD6elem}:
\begin{table}[!hbt]
  \centering
  \begin{eqnarray*}
  \begin{array}{|c|rcl|}
\hline
\mbox{\bf e} & 1_1 &  = & \{x,y,z\} \\
\hline \null & 2_1 &  = & \left\{\frac{1}{2} \left(x+\sqrt{3}
   y\right),\frac{1}{2} \left(y-\sqrt{3} x\right),z\right\} \\
A & 2_2 &  = & \left\{\frac{1}{2} \left(x-\sqrt{3}
   y\right),\frac{1}{2} \left(\sqrt{3} x+y\right),z\right\} \\
\hline \null & 3_1 &  = & \left\{\frac{1}{2} \left(\sqrt{3}
   y-x\right),\frac{1}{2} \left(-\sqrt{3} x-y\right),z\right\}
   \\
A^2 & 3_2 &  = & \left\{\frac{1}{2} \left(-x-\sqrt{3}
   y\right),\frac{1}{2} \left(\sqrt{3} x-y\right),z\right\} \\
 \hline
A^3 & 4_1 &  = & \{-x,-y,z\} \\
 \hline
\null & 5_1 &  = & \{-x,y,-z\} \\
B & 5_2 &  = & \left\{\frac{1}{2} \left(x-\sqrt{3}
   y\right),\frac{1}{2} \left(-\sqrt{3} x-y\right),-z\right\}
   \\
\null & 5_3 &  = & \left\{\frac{1}{2} \left(x+\sqrt{3}
   y\right),\frac{1}{2} \left(\sqrt{3} x-y\right),-z\right\} \\
 \hline
\null & 6_1 &  = & \left\{\frac{1}{2} \left(-x-\sqrt{3}
   y\right),\frac{1}{2} \left(y-\sqrt{3} x\right),-z\right\} \\
BA & 6_2 &  = & \{x,-y,-z\} \\
\null & 6_3 &  = & \left\{\frac{1}{2} \left(\sqrt{3}
   y-x\right),\frac{1}{2} \left(\sqrt{3} x+y\right),-z\right\}\\
   \hline
\end{array}
\end{eqnarray*}
  \caption{Conjugacy Classes of the Dihedral Group $\mathrm{Dih_6}$}\label{nomiD6elem}
\end{table}
In such a table every group element is uniquely identified by its
action on the three-dimensional vector $\left\{x,y,z\right\}$. The
multiplication table of the group $\mathrm{Dih_6}$ is shown in table
\ref{Dmultab}.
\begin{table}[!hbt]
  \centering
  \begin{eqnarray*}
    \begin{array}{|l|llllllllllll|}
    \hline
 \null &   1_1 & 2_1 & 2_2 & 3_1 & 3_2 & 4_1 & 5_1 & 5_2 & 5_3 & 6_1 &
   6_2 & 6_3 \\
   \hline
 1_1 &   1_1 & 2_1 & 2_2 & 3_1 & 3_2 & 4_1 & 5_1 & 5_2 & 5_3 & 6_1
   & 6_2 & 6_3 \\
 2_1 &   2_1 & 3_1 & 1_1 & 4_1 & 2_2 & 3_2 & 6_3 & 6_1 & 6_2 & 5_1
   & 5_2 & 5_3 \\
 2_2 &   2_2 & 1_1 & 3_2 & 2_1 & 4_1 & 3_1 & 6_1 & 6_2 & 6_3 & 5_2
   & 5_3 & 5_1 \\
 3_1 &   3_1 & 4_1 & 2_1 & 3_2 & 1_1 & 2_2 & 5_3 & 5_1 & 5_2 & 6_3
   & 6_1 & 6_2 \\
 3_2 &   3_2 & 2_2 & 4_1 & 1_1 & 3_1 & 2_1 & 5_2 & 5_3 & 5_1 & 6_2
   & 6_3 & 6_1 \\
 4_1 &   4_1 & 3_2 & 3_1 & 2_2 & 2_1 & 1_1 & 6_2 & 6_3 & 6_1 & 5_3
   & 5_1 & 5_2 \\
 5_1 &   5_1 & 6_1 & 6_3 & 5_2 & 5_3 & 6_2 & 1_1 & 3_1 & 3_2 & 2_1
   & 4_1 & 2_2 \\
 5_2 &   5_2 & 6_2 & 6_1 & 5_3 & 5_1 & 6_3 & 3_2 & 1_1 & 3_1 & 2_2
   & 2_1 & 4_1 \\
 5_3 &   5_3 & 6_3 & 6_2 & 5_1 & 5_2 & 6_1 & 3_1 & 3_2 & 1_1 & 4_1
   & 2_2 & 2_1 \\
 6_1 &   6_1 & 5_2 & 5_1 & 6_2 & 6_3 & 5_3 & 2_2 & 2_1 & 4_1 & 1_1
   & 3_1 & 3_2 \\
 6_2 &   6_2 & 5_3 & 5_2 & 6_3 & 6_1 & 5_1 & 4_1 & 2_2 & 2_1 & 3_2
   & 1_1 & 3_1 \\
 6_3 &   6_3 & 5_1 & 5_3 & 6_1 & 6_2 & 5_2 & 2_1 & 4_1 & 2_2 & 3_1
   & 3_2 & 1_1\\
   \hline
\end{array}
\end{eqnarray*}
  \caption{Multiplication table of the Dihedral Group $\mathrm{Dih_6}$}\label{Dmultab}
\end{table}
\subsection{Irreducible representations of the dihedral group $\mathrm{D_6}$ and the character table}
The group $\mathrm{D_6}$ has six conjugacy classes. Therefore
according to theory we expect six irreducible representations that
we name $D_i$, $i\, =\, 1,\dots, 6$. Let us briefly describe them.
The first four representations are one-dimensional.
\subsubsection{$D_1$ : the identity representation}
The identity representation which exists for all groups is that one
where to each element of $\mathrm{D_6}$ we associate the number $1$
\begin{equation}\label{D6identD1}
    \forall \, \gamma \, \in \, \mathrm{O} \,\, : \quad D_1(\gamma) \, = \, 1
\end{equation}
Obviously the character of such a representation is:
\begin{equation}\label{caretterusD6D1}
    \chi_1 \, = \, \{1,1,1,1,1\}
\end{equation}
\subsubsection{$D_2$ : the second one-dimensional representation}
The representation $D_2$ is also one-dimensional. It is constructed
as follows.
\vfill
\newpage
\begin{equation}
\begin{array}{ccccccc}
 \forall \, \gamma & \in & \{\mbox{\bf e}\} & : & D_2(\gamma) & = & 1 \\
  \forall \, \gamma & \in & \{A\} & : & D_2(\gamma) & = & -1 \\
   \forall \, \gamma & \in &\left\{A^2\right\}& : & D_2(\gamma) & = & 1 \\
 \forall \, \gamma & \in  & \left\{A^3\right\}& : & D_2(\gamma) & = & -1 \\
   \forall \, \gamma & \in & \{B\}& : & D_2(\gamma) & = & 1 \\
  \forall \, \gamma & \in  & \{{BA}\} & : & D_2(\gamma) & = & -1 \\
\end{array}
\end{equation}
Clearly the corresponding character vector is the following one.
\begin{equation}\label{caretterusD6D2}
    \chi_2 \, = \, \{1,-1,1,-1,1,-1\}
\end{equation}
Said in another way, this is the representation where $A \, = \, -1$
and $B\, = \,1$.
\subsubsection{$D_3$ : the third one-dimensional representation}
The representation $D_3$ is also one-dimensional. It is constructed
as follows.
\begin{equation}
\begin{array}{ccccccc}
 \forall \, \gamma & \in & \{\mbox{\bf e}\} & : & D_2(\gamma) & = & 1 \\
  \forall \, \gamma & \in & \{A\} & : & D_2(\gamma) & = & -1 \\
   \forall \, \gamma & \in &\left\{A^2\right\}& : & D_2(\gamma) & = & 1 \\
 \forall \, \gamma & \in  & \left\{A^3\right\}& : & D_2(\gamma) & = & -1 \\
   \forall \, \gamma & \in & \{B\}& : & D_2(\gamma) & = & -1 \\
  \forall \, \gamma & \in  & \{{BA}\} & : & D_2(\gamma) & = & 1 \\
\end{array}
\end{equation}
Clearly the corresponding character vector is the following one.
\begin{equation}\label{caretterusD6D3}
    \chi_3 \, = \, \{1,-1,1,-1,-1,1\}
\end{equation}
Said in another way, this is the representation where $A \, = \, -1$
and $B\, = \, -1$.
\subsubsection{$D_4$ : the fourth one-dimensional representation}
The representation $D_4$ is also one-dimensional. It is constructed
as follows.
\begin{equation}
\begin{array}{ccccccc}
 \forall \, \gamma & \in & \{\mbox{\bf e}\} & : & D_2(\gamma) & = & 1 \\
  \forall \, \gamma & \in & \{A\} & : & D_2(\gamma) & = & 1 \\
   \forall \, \gamma & \in &\left\{A^2\right\}& : & D_2(\gamma) & = & 1 \\
 \forall \, \gamma & \in  & \left\{A^3\right\}& : & D_2(\gamma) & = & 1 \\
   \forall \, \gamma & \in & \{B\}& : & D_2(\gamma) & = & -1 \\
  \forall \, \gamma & \in  & \{{BA}\} & : & D_2(\gamma) & = & -1 \\
\end{array}
\end{equation}
Clearly the corresponding character vector is the following one.
\begin{equation}\label{caretterusD6D2bis}
    \chi_4 \, = \, \{1,1,1,1,-1,-1\}
\end{equation}
Said in another way, this is the representation where $A \, = \, 1$
and $B\, = \, -1$.
\subsubsection{$D_5$ : the first two-dimensional representation}
The representation $D_5$ is two-dimensional and it corresponds to a
homomorphism:
\begin{equation}\label{D6mapD5}
    D_5 \, : \quad \mathrm{Dih_6} \, \rightarrow \, \mathrm{SL(2,\mathbb{C})}
\end{equation}
which associates to each element of the dihedral group a $2 \times
2$ complex valued matrix of determinant one. The homomorphism is
completely specified by giving the two matrices representing the two
generators:
\begin{equation}\label{D6mapD5bis}
    D_5(A) \, = \,
\left(
\begin{array}{ll}
 e^{\frac{i \pi }{3}} & 0 \\
 0 & e^{-\frac{i \pi }{3}}
\end{array}
\right)\quad ; \quad D_5(B) \, = \, \left(
\begin{array}{ll}
 0 & 1 \\
 1 & 0
\end{array}
\right)
\end{equation}
The character vector of $D_5$ is easily calculated from the above
information and we have:
\begin{equation}\label{caretterusD6D5}
    \chi_5 \, = \, \{2 , \, 1 ,\, -1 ,\, -2,\, 0,\,  0\}
\end{equation}
\subsubsection{$D_6$ :  the second two-dimensional representation}
The representation $D_6$ is also two-dimensional and it corresponds
to a homomorphism:
\begin{equation}\label{D6mapD5tris}
    D_6 \, : \quad \mathrm{Dih_6} \, \rightarrow \, \mathrm{SL(2,\mathbb{C})}
\end{equation}
which associates to each element of the dihedral group a $2 \times
2$ complex valued matrix of determinant one. The homomorphism is
completely specified by giving the two matrices representing the two
generators:
\begin{equation}\label{D6mapD6}
    D_6(A) \, = \,
\left(
\begin{array}{ll}
 e^{\frac{2 i \pi }{3}} & 0 \\
 0 & e^{-\frac{2 i \pi }{3}}
\end{array}
\right)\quad ; \quad D_6(B) \, = \, \left(
\begin{array}{ll}
 0 & 1 \\
 1 & 0
\end{array}
\right)
\end{equation}
The character vector of $D_6$ is easily calculated from the above
information and we have:
\begin{equation}\label{caretterusD6D6}
    \chi_6 \, = \, \{2 , \, -1 ,\, -1 ,\, 2,\, 0,\,  0\}
\end{equation}
\begin{table}[!hbt]
  \centering
  \begin{eqnarray*}
    \begin{array}{||l||cccccc||}
    \hline
    \hline
{\begin{array}{cc} \null &\mbox{Class}\\
\mbox{Irrep} & \null\\
\end{array}}& \{\mbox{\bf e},1\} & \{A,2\} & \left\{A^2,2\right\} & \left\{A^3,1\right\} & \{B,3\} &\{{BA},3\} \\
\hline \hline
 D_1 \, , \quad \chi_1 \, = \,& 1 & 1 & 1 & 1 & 1 & 1 \\
 \hline
 D_2 \, , \quad \chi_2 \, = \,& 1 & -1 & 1 & -1 & 1 & -1 \\
 \hline
 D_3 \, , \quad \chi_3 \, = \,& 1 & -1 & 1 & -1 & -1 & 1 \\
 \hline
 D_4 \, , \quad \chi_4 \, = \,& 1 & 1 & 1 & 1 & -1 & -1 \\
 \hline
 D_5 \, , \quad \chi_5 \, = \,& 2 & 1 & -1 & -2 & 0 & 0 \\
 \hline
 D_6\, , \quad \chi_6 \, = \,& 2 & -1 & -1 & 2 & 0 & 0\\
 \hline
\end{array}
\end{eqnarray*}
  \caption{The character table of the dihedral group $\mathrm{Dih_6}$}\label{D6caratter}
\end{table}
The character table of the $\mathrm{Dih_6}$ group is summarized in
table \ref{D6caratter}.
\section{Extensions of the Point Group with translations and the Universal Classifying Group}
\label{maingruppo} We come now to what constitutes the main
mathematical point of \cite{Fre:2015mla}, namely the extension of
the point group with appropriate discrete subgroups of the
compactified translation group $\mathrm{U(1)}^3$. This issue bears
on a classical topic dating back to the XIX century, which was
developed by crystallographers and in particular by the great
russian mathematician Fyodorov \cite{fyodorovcryst}. We refer here
to the issue of space groups which historically resulted into the
classification of the $230$ crystallographic groups, well known in
the chemical literature, for which an international system of
notations and conventions was established that is available in
numerous  encyclopedic tables and books. Although in
\cite{Fre:2015mla}  one key-point of the logic that leads to  the
classification of space groups, was utilized, yet the pursued goal
happened to be slightly different. Indeed what was aimed at was not
 the identification of the various space groups, rather the construction
of what was christened in \cite{Fre:2015mla} the \textit{Universal
Classifying Group}, namely  a single large group which contains all
the existing \textit{space groups} as subgroups. It was advocated in
\cite{Fre:2015mla} that such   \textit{Universal Classifying Group}
is the one appropriate to organize the eigenfunctions of the
$\star_g \mathrm{d}$-operator into irreducible representations  and
eventually to uncover the available hidden symmetries of all
Arnold-Beltrami flows.
\subsection{Group extensions}
\label{spaziogruppi} The idea of space groups is naturally related
with the notion of group-extensions. Here we analyze how it arises.
The covering manifold of the $T^3$ torus is $\mathbb{R}^3$ which can
be regarded as the following coset manifold:
\begin{equation}\label{fantacosetto}
\mathbb{R}^3 \, \simeq \, \frac{\mathbb{E}^3}{\mathrm{SO(3)}} \quad
; \quad  \, \mathbb{E}^3 \, \equiv \, \mathrm{ISO(3)} \, \doteq \,
\mathcal{T}^3 \ltimes \mathrm{SO(3)}
\end{equation}
where $\mathcal{T}^3$  is the three dimensional translation group
acting on $\mathbb{R}^3$ in the standard way:
\begin{equation}\label{trasluco}
\forall \mathbf{t} \, \in \, \mathcal{T}^3 \, ,\, \forall \mathbf{x}
\, \in \, \mathbb{R}^3\quad | \quad\mathbf{t} \, : \,   \mathbf{x}
\, \rightarrow \, \mathbf{x}\, + \, \mathbf{t}
\end{equation}
and the Euclidian group $\mathbb{E}^3$ is the semi-direct product of
the  translation group $\mathcal{T}^3$ with the proper rotation
group $\mathrm{SO(3)}$.
\par
In an abstract notation the semi-direct product of  two groups
$\mathrm{T}$ and $\mathrm{G_0}$, where $\mathrm{T}$ is abelian and
supports an action of $\mathrm{G_0}$ which is not necessarily
abelian:
\begin{equation}\label{astrattone}
    \forall \gamma \in \mathrm{G_0}\quad \text{and} \quad  \forall t \in \mathrm{T}
    \quad\quad\quad  \gamma \quad : \quad \mathrm{T} \, \longrightarrow \,\mathrm{T}
    \, \quad ; \quad
    \gamma\circ t \,\in \, T
\end{equation}
can be presented as it follows. As a set the semidirect product:
\begin{equation}\label{strggoG}
   \mathrm{G} \, = \,\mathrm{T} \, \ltimes \, \mathrm{G_0}
\end{equation}
is the cartesian product $\mathrm{T} \times \mathrm{G_0}$ and the
product law $\bullet$ on the set of pairs of elements $\left(t\in
\mathrm{T}\, , \, \gamma \in \mathrm{G_0} \right)$ is the following
one:
\begin{equation}\label{diretprodut}
    \left(t\, , \, \alpha\right)\,\bullet\,\left(w\, , \,
    \beta\right)\, = \, \left( t+\alpha\circ w \, , \, \alpha \cdot
    \beta\right)
\end{equation}
where the product operation for the abelian group $\mathrm{T}$ has
been denoted with $+$, (the inverse is $-$) and the neutral element
is $0$, while for the group $\mathrm{\mathrm{G_0}}$ the product
operation is denoted by $\cdot$ and the neutral element is denoted
by $\mathbf{1}$. As a consequence of the definition of direct
product the original abelian group $\mathrm{T}$ is a normal subgroup
of $\mathrm{G}$:
\begin{equation}\label{fradiacono}
    \mathrm{T} \, \lhd \,\mathrm{G}
\end{equation}
The direct product construction is an example of the realization of
the following exact sequence of four maps $\mu_i$:
\begin{equation}\label{carnevalepisano}
    0 \,\underbrace{\stackrel{\iota}{\longrightarrow}}_{\mu_1} \, \mathrm{T}
    \,\underbrace{\stackrel{\iota}{\longrightarrow}}_{\mu_2}
    \, \mathrm{G} \, \,\underbrace{\stackrel{\pi}{\longrightarrow}}_{\mu_3} \,
    \mathrm{\mathrm{G_0}} \,\underbrace{\stackrel{\pi}{\longrightarrow}}_{\mu_4} \,\mathbf{ 1}
\end{equation}
The first map is the injection map of the neutral element of
$\mathrm{T}$ into the group it pertains to. The second map is the
injection map of the abstract group $\mathrm{T}$ as a normal
subgroup in some group $\mathrm{G}$, the third map is the projection
onto the quotient $\mathrm{\mathrm{G_0}} \, \equiv \,
\frac{\mathrm{G}}{\mathrm{T}}$, the fourth map is the projection of
the entire $\mathrm{\mathrm{G_0}}$ onto its neutral element
$\mathbf{1}$. The exactness property of the sequence:
\begin{equation}\label{crinaldo}
    \text{ker}(\mu_i) \, = \, \text{Im}(\mu_{i+1})
\end{equation}
is evident from the description. Any time we succeed in realizing
the middle term $\mathrm{G}$ in such an exact sequence as that in
eq. (\ref{carnevalepisano}) we say that $\mathrm{G}$ is a
\textbf{group extension} of $\mathrm{T}$ by means of the group
$\mathrm{G_0}$ which is supposed to have an automorphic action on
$\mathrm{T}$. The direct product is just one example of the
realizations of such group extensions but it is not the only one.
\subsubsection{The exact sequence for space groups and the inhomogeneous group
$\mathfrak{Ip}_\Lambda$} In modern mathematical language the space
groups of crystallography emerge just in the  way described above.
We choose a crystallographic lattice $\Lambda$ and a finite point
group $\mathfrak{P}\subset \mathrm{SO(3)}$ that is the maximal one
$\mathfrak{P}_{\Lambda}^{max}$ leaving $\Lambda$ invariant or one of
its subgroups and we write the exact sequence
\begin{equation}\label{carnevalelivorno}
    0 \,\stackrel{\iota}{\longrightarrow} \, \mathrm{\Lambda}
    \,\stackrel{\iota}{\longrightarrow}
    \, \mathfrak{S} \, \,\stackrel{\pi}{\longrightarrow} \,
    \mathrm{\mathfrak{P}} \,\stackrel{\pi}{\longrightarrow}\,\mathbf{ 1}
\end{equation}
where $\mathfrak{S}$ is the space group. One possible construction
of the exact sequence is the already mentioned semi-direct product:
\begin{equation}\label{ltimo}
    \mathfrak{S}_\times \, = \, \Lambda \,\ltimes \,\mathfrak{P}
\end{equation}
which we can reproduce quite conveniently through the use of
$4\times 4$ matrices of the following type:
\begin{equation}\label{quattroruote}
\forall \left(\mathfrak{t} , \, \gamma\right) \, \in \,
\mathfrak{S}_{\times} \, \rightarrow\, {D}_\times\left[
\left(\mathfrak{t} , \, \gamma\right) \right] \, = \, \left(
\begin{array}{c|c}
\hat{\gamma} & {\mathfrak{t}} \\
\hline 0& 1
\end{array}
\right)
\end{equation}
where by $\hat{\gamma}$ we mean the $3\times 3$ matrix realization
of the abstract group element $\gamma$ in the defining
representation of $\mathfrak{P}\subset \mathrm{SO(3)}$. Let us see
how we can realize the exact sequence (\ref{carnevalelivorno}) in a
more general way.
\par
We begin by observing that harmonic analysis on $\mathbb{R}^3$ is a
complicated matter of functional analysis since $\mathcal{T}^3$ is a
non-compact group and its unitary irreducible representations are
infinite-dimensional. The landscape changes drastically when we
compactify our manifold from $\mathbb{R}^3$ to the three torus
$T^3$. Compactification is obtained taking the  quotient of
$\mathbb{R}^3$ with respect to the lattice $\Lambda \, \subset \,
\mathcal{T}^3$. As a result of this quotient the manifold becomes
$\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ but also the
isometry group is reduced. Instead of $\mathrm{SO(3)}$ as rotation
group we are left with its discrete subgroup
$\mathfrak{P}^{max}_\Lambda \, \subset \, \mathrm{SO(3)}$  which
maps the lattice $\Lambda$ into itself (the maximal point group or a
subgroup thereof $\mathfrak{P}_\Lambda \subset
\mathfrak{P}^{max}_\Lambda \, \subset \, \mathrm{SO(3)}$) and
instead of the translation subgroup $\mathcal{T}^3$ we are left with
the quotient group:
\begin{equation}\label{quozienTra}
    \mathfrak{T}^3_\Lambda \, \equiv \, \frac{\mathcal{T}^3}{\Lambda} \, \simeq \,
    \mathrm{U(1)} \times \mathrm{U(1) }\times \mathrm{U(1)}
\end{equation}
In this way we obtain a new group which replaces the Euclidian group
and which is the semidirect product of $\mathfrak{T}^3_\Lambda $
with the point group $\mathfrak{P}_\Lambda$  :
\begin{equation}\label{goticoIG}
    \mathfrak{Ip}_{\Lambda}  \, \equiv \,   \mathfrak{T}^3_\Lambda \,\ltimes
    \, \mathfrak{P}_\Lambda
\end{equation}
The group $ \mathfrak{Ip}_{\Lambda}$ that can be named the
\textbf{Inhomogeneous Point Group} is an exact symmetry of Beltrami
equation (\ref{formaduale}) and its action is naturally defined on
the parameter space of any of its solutions $\mathbf{V}\left(
{\mathbf{x}} | \mathbf{F}\right)$ that we can obtain by means of the
algorithm  described in section \ref{algoritmo}.  To appreciate this
point let us state that every component of the vector field
$\mathbf{V}\left( {\mathbf{x}} | \mathbf{F}\right)$ associated with
a $\mathfrak{P}_\Lambda$ point--orbit $\mathcal{O}$ is a linear
combinations of the functions $\cos\left[2\pi \, \mathbf{k}_i \cdot
{\mathbf{x}}\right]$ and $\sin\left[2\pi \, \mathbf{k}_i \cdot
{\mathbf{x}}\right]$, where $\mathbf{k}_i \, \in \, \mathcal{O}$ are
all the momentum vectors contained in the orbit. Consider next the
same functions in a translated point of the three torus
${\mathbf{x}}^\prime \, = \,{\mathbf{x}}\,  + \, \mathbf{c}$ where
$\mathbf{c} \, = \, \{\xi_1\,,\,\xi_2\, , \, \xi_3\,\}$ is a
representative of an equivalence class $\mathfrak{c}$ of constant
vectors defined modulo the lattice:
\begin{equation}\label{cribulla}
\mathfrak{c} \, = \, \mathbf{c} \, + \, \mathfrak{t} \quad ; \quad
\forall \mathfrak{t}\, \in \, \Lambda
\end{equation}
The above equivalence classes are the elements  of the quotient
group $\mathfrak{T}^3_\Lambda $. Using standard trigonometric
identities $\cos\left[2\pi \, \mathbf{k}_i \cdot {\mathbf{x}}\, + \,
2\pi \, \mathbf{k}_i \cdot \mathbf{c} \right]$  can be reexpressed
as a linear combination of the $\cos\left[2\pi \, \mathbf{k}_i \cdot
{\mathbf{x}}\right]$ and $\sin\left[2\pi \, \mathbf{k}_i \cdot
{\mathbf{x}}\right]$ functions with coefficients that depend on
trigonometric functions of $\mathfrak{c}$. The same is true of
$\sin\left[2\pi \, \mathbf{k}_i \cdot {\mathbf{x}}\, + \, 2\pi \,
\mathbf{k}_i \cdot \mathbf{c} \right]$. Note also that because of
the periodicity of the trigonometric functions, the shift in their
argument by a lattice translation is not-effective so that one deals
only with the equivalence classes (\ref{cribulla}).  It follows that
for each element $\mathfrak{c}\in \mathfrak{T}^3_\Lambda$  we obtain
a matrix representation $\mathcal{M}_\mathfrak{c}$ realized on the
$F$ parameters and defined by the following equation:
\begin{eqnarray}\label{traslazionaBuh}
   & \mathbf{V}\left({\mathbf{x}}+\mathbf{c} |\mathbf{F}\right)\, =\,
   \mathbf{V}\left({\mathbf{x}} |\mathcal{M}_\mathfrak{c} \mathbf{F}\right) &
\end{eqnarray}
As we already noted in eq.(\ref{Rtrasformogen}), for any group
element $\gamma \, \in \, \mathfrak{P}_\Lambda$ we also have a
matrix representation induced on the parameter space by the same
mechanism:
\begin{equation}\label{RotazioneBuh}
\forall \, \gamma \, \in \, \mathfrak{P}_\Lambda \, : \quad
\gamma^{-1} \,\cdot \, \mathbf{V}\left(\gamma \,\cdot \,
{\mathbf{x}}\, | \, \mathbf{F}\right) \, = \, \mathbf{V}\left(
{\mathbf{x}}\, | \, \mathfrak{R}[\gamma] \,\cdot \,
\mathbf{F}\right)
\end{equation}
Combining eq.s(\ref{traslazionaBuh}) and (\ref{RotazioneBuh}) we
obtain a matrix realization of the entire group
$\mathfrak{G}_{\Lambda} $ in the following way:
\begin{eqnarray}
\mathbf{V}\left(\gamma\, \cdot \,  {\mathbf{x}}+\mathbf{c}
|\mathbf{F}\right)&=& \gamma \, \cdot \, \mathbf{V}\left(
{\mathbf{x}}
\,  |\, \mathfrak{R}[\gamma] \cdot \mathcal{M}_\mathfrak{c} \,\cdot \,\mathbf{F}\right) \\
&\Downarrow&\nonumber\\
\forall \left(\gamma \, , \, \mathfrak{c}\right) \, \in \,
\mathfrak{Ip}_{\Lambda} &\rightarrow& D\left[ \left(\gamma \, , \,
\mathfrak{c}\right) \right] \, = \, \mathfrak{R}[\gamma] \cdot
\mathcal{M}_\mathfrak{c}\label{direttocoriandolo}
\end{eqnarray}
Actually the construction of Beltrami vector fields in the lowest
lying point-orbit, which usually yields a faithful matrix
representation of all group elements, can be regarded as an
automatic way of taking  the quotient (\ref{quozienTra}) and the
resulting representation can be considered the defining
representation of the group $\mathfrak{Ip}_{\Lambda}$.
\par
The next point in the logic which leads to space groups is the
following observation. $\mathfrak{Ip}_{\Lambda}$ is an unusual
mixture of a discrete group (the point group ) with a continuous one
(the translation subgroup $\mathfrak{T}^3_\Lambda $). This latter is
rather trivial, since its action corresponds to shifting the origin
of coordinates in three-dimensional space and, from the point of
view of the first order differential system that defines
trajectories (see eq.(\ref{streamlines})), it simply corresponds to
varying the integration constants. Yet there are in
$\mathfrak{Ip}_{\Lambda}$ some discrete subgroups which can be
isomorphic to the point group  $\mathfrak{P}_{\Lambda}$, or to one
of its subgroups $\mathfrak{H}_\Lambda \, \subset \,
\mathfrak{P}_{\Lambda}$, without being their conjugate in
$\mathfrak{Ip}_{\Lambda}$. Such groups cannot be disposed of by
shifting the origin of coordinates and consequently they can encode
non-trivial hidden symmetries of the dynamical system
(\ref{streamlines}).
\par
The precise mathematical way of thinking is encoded in the already
presented exact sequence (\ref{carnevalelivorno}). Given the point
group $\mathfrak{P}_\Lambda$ and its semidirect product extension
with translations reduced to the unit cell $\mathcal{T}^3_{unit}
\simeq \mathrm{U(1)\times U(1) \times U(1)} $, namely
$\mathfrak{G}_{\Lambda} $, the original point group can be
identified as the quotient group:
\begin{equation}\label{colbacco}
    \mathfrak{P}_\Lambda \, \simeq \,
    \frac{\mathfrak{Ip}_{\Lambda}}{\mathcal{T}^3_{unit}}
\end{equation}
since $\mathcal{T}^3_{unit}$ is a normal subgroup:
\begin{equation}\label{cappelloconorecchie}
    \mathcal{T}^3_{unit} \lhd \mathfrak{Ip}_{\Lambda}
\end{equation}
We would like to construct the entire equivalence class of elements
in $ \mathfrak{Ip}_{\Lambda}$ for each element $\gamma \in
\mathfrak{P}_\Lambda$. Choosing representatives in these classes we
can realize the various group extensions $\mathfrak{S}$ that can
occupy the middle point in the exact sequence
(\ref{carnevalelivorno}).
\par
This is the mission accomplished by crystallographers the result of
the mission being the classification of space groups. It suffices to
realize a generalized copy of each generator of the point group and
by means of multiplication we obtain the equivalence classes of each
point group element.
\par
This leads to the so named Frobenius congruences
\cite{Aroyo}\cite{Souvignier}. Let us outline this construction.
\subsubsection{Frobenius congruences} Following classical
approaches we use the already introduced $4\times 4$ matrix
representation of the group $\mathfrak{Ip}_{\Lambda}$. Performing
the matrix product of two elements, in the translation block one has
to take into account equivalence modulo lattice $\Lambda$, namely
\begin{equation}\label{moduloLatte}
\left( \begin{array}{c|c}
\gamma_1 & \mathfrak{c_1} \\
\hline 0& 1
\end{array}
\right) \, \cdot \, \left( \begin{array}{c|c}
\gamma_2 & \mathfrak{c_2} \\
\hline 0& 1
\end{array}
\right) \, = \,  \left( \begin{array}{c|c}
\gamma_1\, \cdot \, \gamma_2 & \gamma_1\,  \mathfrak{c_2} \,
+ \, \mathfrak{c_1}\, + \, \Lambda \\
\hline 0& 1
\end{array}
\right)
\end{equation}
Utilizing this notation the next step consists of introducing
translation deformations of the generators of the point group
$\mathfrak{P}_\Lambda$ searching for deformations that cannot be
eliminated by conjugation with elements of the normal subgroup
$\mathfrak{T}^3 \lhd \mathfrak{Ip}_\Lambda$. We go through the steps
of such a construction both in the case of the maximal point group
for the cubic lattice $\mathfrak{P}_{cubic}^{max}\,= \,
\mathrm{O_{24}}$, denoting with $\mathrm{O_{24}}$ the octahedral
group, and in the case of the maximal point group for the
alternative hexagonal lattice $\mathfrak{P}_{hexag}^{max}\,= \,
\mathrm{Dih_6}$.
\subsubsection{Frobenius congruences for the
Octahedral Group $\mathrm{O_{24}}$} The octahedral  group is
abstractly defined by the presentation displayed in
eq.(\ref{octapresa}). As a first step we parameterize the candidate
deformations of the two generators $T$ and $S$ in the following way:
\begin{equation}\label{deformuccia}
\hat{T} \, = \, \left(
\begin{array}{lll|l}
 0 & 1 & 0 & \tau _1 \\
 0 & 0 & 1 & \tau _2 \\
 1 & 0 & 0 & \tau _3 \\
 \hline
 0 & 0 & 0 & 1
\end{array}
\right) \quad ; \quad \hat{S} \, = \, \left(
\begin{array}{lll|l}
 0 & 0 & 1 & \sigma _1 \\
 0 & -1 & 0 & \sigma _2 \\
 1 & 0 & 0 & \sigma _3 \\
 \hline
 0 & 0 & 0 & 1
\end{array}
\right)
\end{equation}
which should be compared with eq.(\ref{generatiTS}). Next we try
impose on the deformed generators the defining relations of
$\mathrm{O}_{24}$. By explicit calculation we find:
\begin{eqnarray}\label{finocchiona}
 \hat{T}^3 & = &    \left(
\begin{array}{lll|l}
 1 & 0 & 0 & \tau _1+\tau _2+\tau _3 \\
 0 & 1 & 0 & \tau _1+\tau _2+\tau _3 \\
 0 & 0 & 1 & \tau _1+\tau _2+\tau _3 \\
 \hline
 0 & 0 & 0 & 1
\end{array}
\right) \quad ; \quad \hat{S}^2 \, = \, \left(
\begin{array}{lll|l}
 1 & 0 & 0 & \sigma _1+\sigma _3 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & \sigma _1+\sigma _3 \\
 \hline
 0 & 0 & 0 & 1
\end{array}
\right) \nonumber\\
\left(\hat{S}\hat{T}\right)^4 & = & \left(
\begin{array}{lll|l}
 1 & 0 & 0 & 4 \sigma _1+4 \tau _3 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 \hline
 0 & 0 & 0 & 1
\end{array}
\right)
\end{eqnarray}
so that we obtain the conditions:
\begin{equation}\label{frobeniale}
\tau _1+\tau _2+\tau _3  \, \in \, \mathbb{Z}\quad ; \quad \sigma
_1+\sigma _3 \, \in \, \mathbb{Z} \quad ; \quad 4 \sigma _1+4 \tau
_3 \, \in \, \mathbb{Z}
\end{equation}
which are the Frobenius congruences for the present case. Next we
consider the effect of conjugation with the most general translation
element of the group $\mathfrak{T}^3\lhd \mathfrak{Ip}_{cubic}$.
Just for convenience we parameterize the translation subgroup as
follows:
\begin{equation}\label{tmatto}
    \mathfrak{t} \, = \, \left(
\begin{array}{lll|l}
 1 & 0 & 0 & a+c \\
 0 & 1 & 0 & b \\
 0 & 0 & 1 & a-c \\
 \hline
 0 & 0 & 0 & 1
\end{array}
\right)
\end{equation}
and we get:
\begin{equation}\label{giambatto}
    \mathfrak{t} \, \hat{T} \, \mathfrak{t}^{-1} \, = \, \left(
\begin{array}{lll|l}
 0 & 1 & 0 & a-b+c+\tau _1 \\
 0 & 0 & 1 & -a+b+c+\tau _2 \\
 1 & 0 & 0 & \tau _3-2 c \\
 \hline
 0 & 0 & 0 & 1
\end{array}
\right)\quad ; \quad \mathfrak{t} \, \hat{S} \, \mathfrak{t}^{-1} \,
= \,\left(
\begin{array}{lll|l}
 0 & 0 & 1 & 2 c+\sigma _1 \\
 0 & -1 & 0 & 2 b+\sigma _2 \\
 1 & 0 & 0 & \sigma _3-2 c \\
 \hline
 0 & 0 & 0 & 1
\end{array}
\right)
\end{equation}
This shows that by using the parameters $b,c$ we can always put
$\sigma_1 \, = \, \sigma_2 \, = \,0$, while using the parameter $a$
we can put $\tau_1 \, = \,0$ (this is obviously only one possible
gauge choice, yet it is the most convenient) so that Frobenius
congruences reduce to:
\begin{equation}\label{calugone}
     \tau _2+\tau _3  \, \in \, \mathbb{Z}\quad ; \quad \sigma _3 \,
      \in \, \mathbb{Z} \quad ; \quad 4 \tau _3 \, \in \, \mathbb{Z}
\end{equation}
Eq.(\ref{calugone}) is of great momentum. It tells us that any non
trivial subgroup $\hat{\mathfrak{P}}\subset \mathfrak{Ip}_{cubic}$
which is isomorphic to the point group $\mathrm{O_{24}}$, but not
conjugate to it contains point group elements extended with rational
translations of the form $\mathfrak{c} \, = \, \left\{ \ft{n_1}{4}\,
, \, \ft{n_2}{4}\,  , \, \ft{n_3}{4}\right\}$ with $n_i \in
\mathbb{Z}$.
\paragraph{The example of the group $\mathrm{GS_{24}}$} An example
is provided by the group later named $\mathrm{GS_{24}}$ which will
repeatedly appear in our later discussions of Beltrami solutions. In
the direct product realization of the point group
$\mathfrak{P}=\mathrm{O_{24}}$ the generators $T$ and $S$ were
specified in eq.s (\ref{octapresa}) and (\ref{generatiTS}). In view
of the Frobenius congruences let us set:
\begin{equation}\label{TScap}
    \hat{T} \, = \,\left(
\begin{array}{cccc}
 0 & 0 & 1 & 0 \\
 1 & 0 & 0 & \frac{1}{2} \\
 0 & 1 & 0 & \frac{1}{2} \\
 0 & 0 & 0 & 1 \\
\end{array}
\right) \quad ; \quad \hat{S} \, = \,\left(
\begin{array}{cccc}
 0 & 0 & 1 & \frac{3}{2} \\
 0 & -1 & 0 & \frac{1}{2} \\
 1 & 0 & 0 & \frac{1}{2} \\
 0 & 0 & 0 & 1 \\
\end{array}
\right)
\end{equation}
By an immediate calculation we obtain:
\begin{eqnarray}\label{governolo}
    &&\hat{T}^3 \, = \,\left(
\begin{array}{cccc}
 1 & 0 & 0 & 1 \\
 0 & 1 & 0 & 1 \\
 0 & 0 & 1 & 1 \\
 0 & 0 & 0 & 1 \\
\end{array}
\right) \quad ; \quad \hat{S}^2 \, = \,\left(
\begin{array}{cccc}
 1 & 0 & 0 & 2 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 2 \\
 0 & 0 & 0 & 1 \\
\end{array}
\right)\quad ; \quad
    \left(\hat{S}\cdot\hat{T}\right)^4 \, = \,\left(
\begin{array}{cccc}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 2 \\
 0 & 0 & 0 & 1 \\
\end{array}
\right)\nonumber\\
\end{eqnarray}
The above equation is interpreted by stating that:
\begin{equation}\label{interpretone}
    \hat{T}^3 \, \in \, \Lambda \, \subset \, \mathfrak{S}_{\mathrm{GS}} \quad ;
    \quad \hat{S}^2 \in \, \Lambda \, \subset \, \mathfrak{S}_{\mathrm{GS}}
    \quad ; \quad \left(\hat{S}\cdot\hat{T}\right)^4 \, \in \, \Lambda
    \, \subset \, \mathfrak{S}_{\mathrm{GS}}
\end{equation}
where $\mathfrak{S}_{GS}$ is the space group in the exact sequence:
\begin{equation}\label{carnevaleorbetello}
    0 \,\stackrel{\iota}{\longrightarrow} \, \mathrm{\Lambda}
    \,\stackrel{\iota}{\longrightarrow}
    \, \mathfrak{S}_{\mathrm{GS}} \, \,\stackrel{\pi}{\longrightarrow} \,
    \mathrm{GS_{24}} \,\stackrel{\pi}{\longrightarrow}\,\mathbf{ 1}
\end{equation}
and the lattice normal subgroup is realized within
$\mathfrak{S}_{\mathrm{GS}}$ by all the matrices of the form:
\begin{equation}\label{gringus}
    \mathfrak{S}_{\mathrm{GS}} \, \rhd \, \Lambda \, \ni
    \, \left(
         \begin{array}{ccc|c}
           1 & 0 & 0 & n_1 \\
           0 & 1 & 0 & n_2 \\
           0 & 0 & 1 & n_3 \\
           \hline
           0 & 0 & 0 & 1 \\
         \end{array}
       \right) \quad ; \quad n_i \, \in \, \mathbb{Z}
\end{equation}
The group $\mathrm{GS_{24}}$ is defined as the quotient group:
\begin{equation}\label{divisionescolastica}
    \mathrm{GS_{24}} \, = \,
    \frac{\mathfrak{S}_{\mathrm{GS}}}{\Lambda}\, \sim \mathrm{O_{24}} \, \sim
    \mathrm{S_4}
\end{equation}
and $\mathfrak{S}_{\mathrm{GS}}$ is a group extension of the lattice
group $\Lambda$ by means of the abstract group octahedral point
group $\mathrm{O_{24}} $, yet it is not a semidirect product of the
normal subgroup $\Lambda$ with $\mathrm{O_{24}}$. Indeed the space
group $\mathfrak{S}_{\mathrm{GS}} $ contains translations that do
not belong to the cubic lattice.
\paragraph{A conceptual bifurcation} Up to this point our way and that of crystallographers
was the same: hereafter our paths separate. The crystallographers
classify all possible non trivial groups that extend the point group
with such translation deformations: indeed looking at the
crystallographic tables one realizes that  all known space groups
for the cubic lattice have translation components of the form
$\mathfrak{c} \, = \, \left\{ \ft{n_1}{4}\,  , \, \ft{n_2}{4}\,  ,
\, \ft{n_3}{4}\right\}$. On the other hand, we do something much
simpler which leads to a quite big group containing all possible
Space-Groups as subgroups, together with other subgroups that are
not space groups in the crystallographic sense.
\subsection{The Universal Classifying Group for the cubic lattice:
 $\mathrm{\mathrm{G_{1536}}}$}
\label{universalone} Inspired by the space group  construction and
by Frobenius congruences we just consider the subgroup of
$\mathfrak{G}_{cubic}$ where translations are quantized in units of
$\frac{1}{4}$. In each direction and modulo integers there are just
four translations $0, \, \ft 14, \, \ft 12, \, \ft 34$ so that the
translation subgroup reduces to $\mathbb{Z}_4 \,
\otimes\,\mathbb{Z}_4\, \otimes \, \mathbb{Z}_4$  that has a total
of $64$ elements. In this way we single out a discrete subgroup
$\mathrm{G_{1536}} \, \subset \, \mathfrak{G}_{cubic}$ of order $24
\times 64 \,  =  \, 1536$,  which is simply the semidirect product
of the point group $\mathrm{O_{24}}$ with $\mathbb{Z}_4 \,
\otimes\,\mathbb{Z}_4\, \otimes \, \mathbb{Z}_4$:
\begin{equation}\label{1536defino}
\mathfrak{G}_{cubic} \, \supset \, \mathrm{\mathrm{G_{1536}}} \,
\simeq \, \mathrm{O_{24}} \, \ltimes \, \left (\mathbb{Z}_4 \,
\otimes\,\mathbb{Z}_4\, \otimes \, \mathbb{Z}_4\right)
\end{equation}
We name $\mathrm{\mathrm{G_{1536}}}$ the universal classifying group
of the cubic lattice, and its elements can be labeled as follows:
\begin{equation}\label{elementando1536}
\mathrm{\mathrm{G_{1536}}} \, \in \, \left\{ p_q \, , \, \ft{2
n_1}{4} \, , \, \ft{2 n_2}{4} \, , \, \ft{2 n_3}{4}\right\} \quad
\Rightarrow\quad \left\{ \begin{array}{rcl}
p_q & \in & \mathrm{O_{24}}\\
\left\{ \ft{n_1}{4}\,  , \, \ft{n_2}{4}\,  , \, \ft{n_3}{4}\right\}
& \in & \mathbb{Z}_4 \, \otimes\,\mathbb{Z}_4\, \otimes \,
\mathbb{Z}_4\end{array}\right.
\end{equation}
where for the elements of the point group we use the labels $p_q$
established in eq.(\ref{nomiOelemen}) while for the translation part
our notation encodes an equivalence class of  translation vectors
$\mathfrak{c} \, = \,  \left\{ \ft{n_1}{4}\,  , \, \ft{n_2}{4}\,  ,
\, \ft{n_3}{4}\right\}$. The reason why we use $\left \{\ft{2
n_1}{4} \, , \, \ft{2 n_2}{4} \, , \, \ft{2 n_3}{4}\right\}$ is
simply due to computer convenience. In the quite elaborate
MATHEMATICA codes that were  utilized in \cite{Fre:2015mla} to
derive all the results such a notation was internally used  and the
automatic LaTeX Export of the outputs was  provided in this way. In
view of eq.(\ref{direttocoriandolo}) one can associate an explicit
matrix to each group element of $\mathrm{\mathrm{G_{1536}}}$,
starting from the construction of the Beltrami vector field
associated with one point orbit of the octahedral  group. Then one
can consider such matrices the defining representation of the group
if the representation is faithful. In \cite{Fre:2015mla} the lowest
lying $6$-dimensional orbit  was used which is indeed faithful.
Three matrices are sufficient to characterize completely the
defining representation just as any other representation: the matrix
representing the generator $T$, the matrix representing the
generator $S$ and the matrix representing the translation $\left\{
\ft{n_1}{4}\,  , \, \ft{n_2}{4}\,  , \, \ft{n_3}{4}\right\}$. In
\cite{Fre:2015mla} it was found:
\begin{equation}\label{TSindefi1536}
    \mathfrak{R}^{\mbox{defi}}[T] \, = \, \left(\begin{array}{llllll}
 0 & 0 & 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 0 & 0 & 1 \\
 0 & 1 & 0 & 0 & 0 & 0 \\
 1 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 & 0 & 0
\end{array}\right)\quad ; \quad \mathfrak{R}^{\mbox{defi}}[S]\, = \, \left(
\begin{array}{llllll}
 0 & 0 & 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 0 & 0 & 1 \\
 0 & 1 & 0 & 0 & 0 & 0 \\
 1 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 & 0 & 0
\end{array}
\right)
\end{equation}
\begin{eqnarray}
\mathcal{M}_{\{\ft{2 n_1}{2},\ft{2 n_2}{2},\ft{2
n_3}{2}\}}^{\mbox{defi}} &= & \left(
\begin{array}{llllll}
\cos \left(\ft{\pi}{2}  n _3\right) & 0 &
\sin \left(\ft{\pi}{2}  n _3\right) & 0 & 0 & 0 \\
0 & \cos \left(\ft{\pi}{2}  n _2\right) & 0 & 0 &
   -\sin \left(\ft{\pi}{2}  n _2\right) & 0 \\
 -\sin \left(\ft{\pi}{2}  n _3\right) & 0 & \cos
   \left(\ft{\pi}{2}  n _3\right) & 0 & 0 & 0 \\
 0 & 0 & 0 & \cos \left(\ft{\pi}{2}  n _1\right) & 0 &
   \sin \left(\ft{\pi}{2}  n _1\right) \\
 0 & \sin \left(\ft{\pi}{2}  n _2\right) & 0 & 0 & \cos
   \left(\ft{\pi}{2}  n _2\right) & 0 \\
 0 & 0 & 0 & -\sin \left(\ft{\pi}{2}  n _1\right) & 0 &
   \cos \left(\ft{\pi}{2}  n _1\right)
\end{array}
\right)\nonumber\\
\label{trasladefi1536}
\end{eqnarray}
Relying on the above matrices, any of the 1536 group elements
obtains an explicit $6\times 6$ matrix representation upon use of
formula (\ref{direttocoriandolo}). As already stressed one can
regard that above as the actual definition of the group
$\mathrm{\mathrm{G_{1536}}}$ which from this point on can be studied
intrinsically in terms of pure group theory without any further
reference to lattices, Beltrami flows or dynamical systems.
\subsection{Structure of the $\mathrm{\mathrm{G_{1536}}}$ group
and derivation of its irreps} The identity card of a finite group is
given by the organization of its elements into conjugacy classes,
the list of its irreducible representation and finally its character
table. Since ours is not any of the crystallographic groups, no
explicit information is available in the literature about its
conjugacy classes, its irreps and its character table. Hence the
authors of \cite{Fre:2015mla} were forced to do everything from
scratch by themselves and they could accomplish the task by means of
purposely written MATHEMATICA codes. Most of their results were
presented in the form of tables in the appendices of
\cite{Fre:2015mla}. We will reproduce here those that are most
relevant to the purposes of the present paper referring the reader
to \cite{Fre:2015mla} for additional details.
\paragraph{Conjugacy Classes}
The conjugacy classes of $\mathrm{\mathrm{G_{1536}}}$ are explicitly
presented in appendix A.1 of \cite{Fre:2015mla}. There are 37
conjugacy classes whose populations is distributed as follows:
\begin{description}
  \item[1)]  2 classes of length 1
  \item[2)]  2  classes of length 3
  \item[3)]  2 classes of  length 6
  \item[4)] 1 class of length 8
  \item[5)] 7 classes of length 12
  \item[6)] 4 classes of length 24
  \item[7)] 13 classes of length 48
  \item[8)] 2 classes of length 96
  \item[9)] 4 classes of length 128
\end{description}
It follows that there must be $37$  irreducible representations
whose construction is a task to be solved.
\subsection{Strategy to construct the irreducible representations of a solvable group}
\label{startegos} In general, the derivation of the irreps and of
the ensuing character table of a finite group $G$ is  a quite hard
task. Yet  a definite constructive algorithm can be devised if $G$
is solvable and if one can establish a chain of normal subgroups
ending with an abelian one, whose index is, at each step,  a prime
number $q_i$, namely if we have the following situation:
\begin{eqnarray}\label{parrucchiere}
G &= &   \mathrm{G_{N_p}} \, \rhd \, \mathrm{G_{N_{p-1}}}  \, \rhd \, \dots \,
\rhd \, \mathrm{G_{N_1}}\rhd \, \mathrm{G_{N_0}} \, = \, \mbox{abelian group}
\nonumber\\
&& \left| \frac{\mathrm{G_{N_i}}}{\mathrm{G_{N_{i-1}}}}\right| \, =
\, \frac{\mathrm{N_i}}{\mathrm{N_{i-1}}}\, \equiv \, q_i \, = \,
\mbox{prime integer number}
\end{eqnarray}
The algorithm for the construction of the irreducible
representations is based on an inductive procedure \cite{Aroyo} that
allows to derive the irreps of the group $\mathrm{G_{N_i}}$ if we
know those of the group $\mathrm{G_{N_{i-1}}}$ and if the index
$q_i$ is a prime number. The first step of the induction is
immediately solved because any abelian finite group is necessarily a
direct product of cyclic groups $\mathbb{Z}_k$, whose irreps are all
one dimensional and obtained by assigning to their generator one of
the $k$-th roots of unity. In our case the index $q_i$ is always
either $2$ or $3$  which, to none's  wonder,  is the same situation
met in the construction of  crystallographic group irreps. Hence we
sketch the inductive algorithms with particular reference to the two
cases of $q=2$ and $q=3$.
\subsubsection{The inductive algorithm for irreps}
\label{agoinduzio} To simplify notation we name $\mathcal{G}\, = \,
\mathrm{G_{N_i}}$  and $\mathcal{H}\, = \, \mathrm{G_{N_{i-1}}}$. By
hypothesis $\mathcal{H} \, \lhd \, \mathcal{G}$ is a normal
subgroup. Furthermore $q \, \equiv \, \left
|\frac{\mathcal{G}}{\mathcal{H}} \right| \, = \, $ \textit{prime
number} (in particular $q\, = \, 2, \mbox{or} \, 3$). Let us name
$D_\alpha\left[\mathcal{H},d_\alpha\right]$ the irreducible
representations of the subgroup. The index $\alpha$ (with $\alpha\,
= \, 1,\dots, r_H \, \equiv \,  \# \mbox{ of conj. classes of
$\mathcal{H}$ }$) enumerates them. In each case $d_\alpha$ denotes
the dimension of the corresponding carrying vector space or, in
mathematical jargon, of the corresponding module.
\par
The first step to be taken is to distribute the $\mathcal{H}$ irreps
into conjugation classes with respect to the bigger group.
Conjugation classes of irreps are defined as follows. First one
observes that, given an irreducible representation
$D_\alpha\left[\mathcal{H},d_\alpha\right]$, for every $g \, \in \,
\mathcal{G}$ we can create another irreducible representation
$D^{(g)}_\alpha\left[\mathcal{H},d_\alpha\right]$, named the
conjugate of $D_\alpha\left[\mathcal{H},d_\alpha\right]$  with
respect to $g$. The new representation is as follows:
\begin{equation}\label{Dconiugo}
    \forall \, h \, \in \, \mathcal{H} \, \quad :
    \quad D^{(g)}_\alpha\left[\mathcal{H},d_\alpha\right](h) \, =
    \, D_\alpha\left[\mathcal{H},d_\alpha\right](g^{-1} \, h \, g)
\end{equation}
That the one defined above is a homomorphism of  $\mathcal{H}$ onto
$\mathrm{GL(d_\alpha,\mathbb{R})}$ is obvious and, as a consequence,
it is also obvious that the new representation has the same
dimension as the first. Secondly if $g = \widetilde{h} \, \in \,
\mathcal{H}$ is an element of the subgroup we get:
\begin{equation}\label{maialeH}
\quad D^{(\widetilde{h})}_\alpha\left[\mathcal{H},d_\alpha\right](h) \,
= \, A^{-1} \, D_\alpha\left[\mathcal{H},d_\alpha\right]( h) \, A
\quad \mbox{where} \quad A \, = \,
D_\alpha\left[\mathcal{H},d_\alpha\right](\widetilde{ h})
\end{equation}
so that conjugation amounts simply to a change of basis (a
similarity transformation) inside the same representation. This does
not alter the character vector and the new representation is
equivalent to the old one.  Hence the only non trivial conjugations
to be considered are those with respect to  representatives of the
different equivalence classes in $\frac{\mathcal{G}}{\mathcal{H}}$.
Let us name $\gamma_i $, $(i=0,\dots ,\,q-1)$ a set of
representatives of such equivalence classes and define the orbit of
each irrep $D_\alpha\left[\mathcal{H},d_\alpha\right]$ as it
follows:
\begin{equation}\label{orbetellosulmare}
\mbox{Orbit}_\alpha \, \equiv \, \left\{
D^{(\gamma_0)}_\alpha\left[\mathcal{H},d_\alpha\right] \, , \,
D^{(\gamma_1)}_\alpha\left[\mathcal{H},d_\alpha\right] \, , \, \dots
\, ,
\,D^{(\gamma_{q-1})}_\alpha\left[\mathcal{H},d_\alpha\right]\right\}
\end{equation}
Since the available irreducible representations are a finite set,
every $D^{(\gamma_i)}_\alpha\left[\mathcal{H},d_\alpha\right] $
necessarily is  identified with one of the existing
$D_\beta\left[\mathcal{H},d_\beta\right] $. Furthermore, since
conjugation preserves the dimension, it follows that $d_\alpha \, =
\, d_\beta$. It follows that $\mathcal{H}$-irreps of the same
dimensions $d$  arrange themselves into $\mathcal{G}$-orbits:
\begin{equation}\label{orbetellosulmarebis}
\mbox{Orbit}_\alpha[d] \, = \, \left\{
D_{\alpha_1}\left[\mathcal{H},d\right] \, , \,
D_{\alpha_2}\left[\mathcal{H},d\right] \, , \, \dots \, ,
\,D_{\alpha_q}\left[\mathcal{H},d\right]\right\}
\end{equation}
and there are only two possibilities, either all $\alpha_i \, = \,
\alpha$ are equal (self-conjugate representations) or they are all
different (non conjugate representations).
\par
Once the irreps of $\mathcal{H}$ have been organized into
conjugation orbits, we can proceed to promote them to irreps of the
big group $\mathcal{G}$ according to the following scheme:
\begin{description}
\item[A)] Each self-conjugate $\mathcal{H}$-irrep $D_\alpha\left[\mathcal{H},d\right]$
is uplifted to $q$ distinct irreducible $\mathcal{G}$-representations
of the same dimension $d$, namely $D_{\alpha_{i}}\left[\mathcal{G},d\right]$
where $i\, = \, 1,\dots,q$.
\item[B)] From each orbit $\beta$ of $q$ distinct but conjugate
$\mathcal{H}$-irreps $\left\{ D_{\alpha_1}\left[\mathcal{H},d\right] \, ,
\, D_{\alpha_2}\left[\mathcal{H},d\right] \, , \, \dots \, , \,D_{\alpha_q}\left[\mathcal{H},d\right]\right\}$ one extracts a single ($q\times d$)-dimensional $\mathcal{G}$-representation.
\end{description}
\paragraph{A) Uplifting of self conjugate representations.}
Let $D_{\alpha}\left[\mathcal{H},d\right]$ be a self conjugate irrep.
If the index $q$ of the normal subgroup is a prime number,
this means that $\frac{\mathcal{G}}{\mathcal{H}}\, \simeq \, \mathbb{Z}_q$.
In this case the representatives $\gamma_{j}$ of the $q$ equivalences classes
that form the quotient group can be chosen in the following way:
\begin{equation}\label{gambaletti}
    \gamma_1 \, = \, \mathbf{e} \, , \, \gamma_2 \, = \, g \, , \, \gamma_3
    \, = \, g^2 , \, \dots , \, \gamma_q  \, = \, g^{q-1}
\end{equation}
where $g \, \in \,  \mathcal{G}$ is a single group element
satisfying $g^q \, = \, \mathbf{e}$. The key-point in up-lifting the
representation $D_{\alpha}\left[\mathcal{H},d\right]$ to the bigger
group resides in the determination of a $d\times d$ matrix $U$ that
should satisfy the following constraints:
\begin{eqnarray}
  U^q &=& \mathbf{1} \\
 \forall \, h \, \in \, \mathcal{H} & : &\quad D_{\alpha}\left[\mathcal{H},d\right]
 \left(g^{-1}\, h \, g\right) \,=\, U^{-1}\, D_{\alpha}\left[\mathcal{H},d\right]
 \left(h\right) \, U
\end{eqnarray}
These algebraic equations have exactly $q$ distinct solutions
$U_{[j]}$   and each of the solutions leads to one of the
irreducible $\mathcal{G}$-representations induced by
$D_{\alpha}\left[\mathcal{H},d\right]$. Any element $\gamma \, \in
\, \mathcal{G}$ can be written as $\gamma \, = \, g^p \, h $ with
$p=0,1,\dots ,\, q-1$ and $h \, \in \, \mathcal{H}$. Then it
suffices to write:
\begin{equation}\label{corbezzoli}
    D_{a_j}\left[\mathcal{G},d\right](\gamma) \, = \,  D_{a_j}\left[\mathcal{G},
    d\right]\left (g^p \, h\right) \, = \, U_{[j]}^p \, D_{\alpha}\left[\mathcal{H},d\right] (h)
\end{equation}
\paragraph{B) Uplifting of not self conjugate representations.}
In the case of not self-conjugate representations the induced representation
of dimensions $q\times d$ is constructed relying once again on the possibility
to write all group elements in the form $\gamma \, = \, g^p \, h $
with $p=0,1,\dots ,\, q-1$ and $h \, \in \, \mathcal{H}$.
Furthermore chosen one representation $D_{\alpha}\left[\mathcal{H},d\right]$
in the $q$-orbit
(\ref{orbetellosulmare}), the other members of the orbit can be
represented as $D^{(g^{j})}_\alpha\left[\mathcal{H},d_\alpha\right]$
with $j=1,\,dots ,\, q-1$. In view of this one writes:
\begin{eqnarray}\label{ciccolone}
\forall \, h \, \in \, \mathcal{H} &:&\quad \nonumber\\
D_{\alpha}\left[\mathcal{G},d\right]\left( h \right)&=&
\left(\begin{array}{c|c|c|c|c}
D_{\alpha}\left[\mathcal{H},d\right](h)& 0& 0 &\dots & 0 \\
\hline
0& D^{(g)}_{\alpha}\left[\mathcal{H},d\right](h) &0& \dots & 0 \\
\hline
0 & 0& D^{(g^2)}_{\alpha}\left[\mathcal{H},d\right](h) & \dots & 0 \\
\hline
\vdots & \vdots & \vdots & \vdots &\vdots \\
\hline 0& 0 & \dots & 0 &
D^{(g^{q-1})}_{\alpha}\left[\mathcal{H},d\right](h)
\end{array}\right)\nonumber\\\null &\null & \null \nonumber\\
g &:&\quad D_{\alpha}\left[\mathcal{G},d\right]\left( g \right)\,=\,
\left(\begin{array}{c|c|c|c|c}
0& 0& 0 &\dots & \mathbf{1} \\
\hline
\mathbf{1}& 0 &0& \dots & 0 \\
\hline
0 & \mathbf{1}& 0 & \dots & 0 \\
\hline
\vdots & \vdots & \vdots & \vdots &\vdots \\
\hline 0& 0 & \dots &\mathbf{1} & 0
\end{array}\right)\nonumber\\\null &\null & \null \nonumber\\\gamma \, =
\, g^p \, h  &:&\quad D_{\alpha}\left[\mathcal{G},d\right]\left( g \right)
\,=\, \left(D_{\alpha}\left[\mathcal{G},d\right]\left( g \right) \right)^p
\, D_{\alpha}\left[\mathcal{G},d\right]\left( h \right) \nonumber\\
\end{eqnarray}
\subsection{Derivation of $\mathrm{\mathrm{G_{1536}}}$ irreps}
\label{ciurlacca} Utilizing the above described algorithm,
implemented by means of purposely written MATHEMATICA codes, the
authors of \cite{Fre:2015mla} were able to derive the explicit form
of the $37$ irreducible representations of
$\mathrm{\mathrm{G_{1536}}}$ and its character table. The essential
tool is the following chain of normal subgroups:
\begin{equation}\label{pernicinormaliText}
    \mathrm{\mathrm{G_{1536}} }\, \rhd \, \mathrm{G_{768}} \,
    \rhd \, \mathrm{G_{256}} \, \rhd \, \mathrm{G_{128}} \, \rhd \, \mathrm{G_{64}}
  \end{equation}
where $\mathrm{G_{64}} \,  \sim \, \mathbb{Z}_4 \, \times \,
\mathbb{Z}_4 \, \times \, \mathbb{Z}_4$ is abelian and corresponds
to the compactified translation group. The above chain leads to the
following quotient groups:
\begin{equation}\label{fagianirossiText}
\frac{\mathrm{\mathrm{G_{1536}} }}{\mathrm{G_{768}}} \, \sim \,
\mathbb{Z}_2 \quad ; \quad \frac{\mathrm{G_{768}
}}{\mathrm{G_{256}}} \, \sim \, \mathbb{Z}_3 \quad ; \quad
\frac{\mathrm{G_{256} }}{\mathrm{G_{128}}} \, \sim \, \mathbb{Z}_2
\quad ; \quad \frac{\mathrm{G_{128} }}{\mathrm{G_{64}}} \, \sim \,
\mathbb{Z}_2
\end{equation}
The description of the normal subgroups is given in various sections
of the appendix of \cite{Fre:2015mla}.  The result for the
irreducible representations, thoroughly described also in the
appendix of \cite{Fre:2015mla} is summarized here. The $37$ irreps
are distributed according to the following pattern:
\begin{description}
  \item[a)] 4 irreps of dimension $1$, namely $D_1,\dots,D_4$
  \item[b)] 2 irreps of dimension $2$, namely $D_5,\dots,D_6$
  \item[c)] 12 irreps of dimension $3$, namely $D_6,\dots,D_{18}$
  \item[d)] 10 irreps of dimension $6$, namely $D_7,\dots,D_{28}$
  \item[e)] 3 irreps of dimension $8$, namely $D_{29},\dots,D_{31}$
  \item[f)] 6 irreps of dimension $12$, namely $D_{32},\dots,D_{37}$
\end{description}
The  character table calculated in \cite{Fre:2015mla} is  displayed
here below {\scriptsize
\begin{equation}\label{1charto1536}
\begin{array}{|l|llllllllllllllllll|}
\hline
 0 & C_1 & C_2 & C_3 & C_4 & C_5 & C_6 & C_7 & C_8 & C_9
   & C_{10} & C_{11} & C_{12} & C_{13} & C_{14} & C_{15}
   & C_{16} & C_{17} & C_{18} \\
   \hline
 D_1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1
   & 1 & 1 & 1 & 1 & 1 \\
 D_2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1
   & 1 & 1 & 1 & 1 & 1 \\
 D_3 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & 1 & -1 & 1 & 1 & 1
   & 1 & 1 & -1 & -1 & -1 & -1 \\
 D_4 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & 1 & -1 & 1 & 1 & 1
   & 1 & 1 & -1 & -1 & -1 & -1 \\
 D_5 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2
   & 2 & 2 & 2 & 2 & 2 \\
 D_6 & 2 & 2 & 2 & 2 & -2 & -2 & -2 & 2 & -2 & 2 & 2 & 2
   & 2 & 2 & -2 & -2 & -2 & -2 \\
 D_7 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & -1 & -1 &
   -1 & -1 & -1 & -1 & -1 & -1 \\
 D_8 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & -1 & -1 &
   -1 & -1 & -1 & -1 & -1 & -1 \\
 D_9 & 3 & 3 & 3 & 3 & 1 & 1 & -3 & -1 & 1 & -1 & 3 & 3 &
   -1 & -1 & 1 & 1 & 1 & -3 \\
 D_{10} & 3 & 3 & 3 & 3 & 1 & 1 & -3 & -1 & 1 & -1 & 3 &
   3 & -1 & -1 & 1 & 1 & 1 & -3 \\
 D_{11} & 3 & 3 & 3 & 3 & 1 & 1 & -3 & -1 & 1 & -1 & -1 &
   -1 & 3 & 3 & -3 & 1 & 1 & 1 \\
 D_{12} & 3 & 3 & 3 & 3 & 1 & 1 & -3 & -1 & 1 & -1 & -1 &
   -1 & 3 & 3 & -3 & 1 & 1 & 1 \\
 D_{13} & 3 & 3 & 3 & 3 & -1 & -1 & 3 & -1 & -1 & -1 & 3
   & 3 & -1 & -1 & -1 & -1 & -1 & 3 \\
 D_{14} & 3 & 3 & 3 & 3 & -1 & -1 & 3 & -1 & -1 & -1 & 3
   & 3 & -1 & -1 & -1 & -1 & -1 & 3 \\
 D_{15} & 3 & 3 & 3 & 3 & -1 & -1 & 3 & -1 & -1 & -1 & -1
   & -1 & 3 & 3 & 3 & -1 & -1 & -1 \\
 D_{16} & 3 & 3 & 3 & 3 & -1 & -1 & 3 & -1 & -1 & -1 & -1
   & -1 & 3 & 3 & 3 & -1 & -1 & -1 \\
 D_{17} & 3 & 3 & 3 & 3 & -3 & -3 & -3 & 3 & -3 & 3 & -1
   & -1 & -1 & -1 & 1 & 1 & 1 & 1 \\
 D_{18} & 3 & 3 & 3 & 3 & -3 & -3 & -3 & 3 & -3 & 3 & -1
   & -1 & -1 & -1 & 1 & 1 & 1 & 1 \\
 D_{19} & 6 & 6 & 6 & 6 & 2 & 2 & -6 & -2 & 2 & -2 & -2 &
   -2 & -2 & -2 & 2 & -2 & -2 & 2 \\
 D_{20} & 6 & 6 & 6 & 6 & -2 & -2 & 6 & -2 & -2 & -2 & -2
   & -2 & -2 & -2 & -2 & 2 & 2 & -2 \\
 D_{21} & 6 & -6 & 2 & -2 & 4 & -4 & 0 & 2 & 0 & -2 & 2 &
   -2 & 2 & -2 & 0 & 2 & -2 & 0 \\
 D_{22} & 6 & -6 & 2 & -2 & 4 & -4 & 0 & 2 & 0 & -2 & 2 &
   -2 & 2 & -2 & 0 & 2 & -2 & 0 \\
 D_{23} & 6 & -6 & 2 & -2 & 4 & -4 & 0 & 2 & 0 & -2 & -2
   & 2 & -2 & 2 & 0 & -2 & 2 & 0 \\
 D_{24} & 6 & -6 & 2 & -2 & 4 & -4 & 0 & 2 & 0 & -2 & -2
   & 2 & -2 & 2 & 0 & -2 & 2 & 0 \\
 D_{25} & 6 & -6 & 2 & -2 & -4 & 4 & 0 & 2 & 0 & -2 & 2 &
   -2 & 2 & -2 & 0 & -2 & 2 & 0 \\
 D_{26} & 6 & -6 & 2 & -2 & -4 & 4 & 0 & 2 & 0 & -2 & 2 &
   -2 & 2 & -2 & 0 & -2 & 2 & 0 \\
 D_{27} & 6 & -6 & 2 & -2 & -4 & 4 & 0 & 2 & 0 & -2 & -2
   & 2 & -2 & 2 & 0 & 2 & -2 & 0 \\
 D_{28} & 6 & -6 & 2 & -2 & -4 & 4 & 0 & 2 & 0 & -2 & -2
   & 2 & -2 & 2 & 0 & 2 & -2 & 0 \\
 D_{29} & 8 & -8 & -8 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{30} & 8 & -8 & -8 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{31} & 8 & -8 & -8 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{32} & 12 & 12 & -4 & -4 & 4 & 4 & 0 & 0 & -4 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{33} & 12 & 12 & -4 & -4 & 4 & 4 & 0 & 0 & -4 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{34} & 12 & 12 & -4 & -4 & -4 & -4 & 0 & 0 & 4 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{35} & 12 & 12 & -4 & -4 & -4 & -4 & 0 & 0 & 4 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{36} & 12 & -12 & 4 & -4 & 0 & 0 & 0 & -4 & 0 & 4 & 4
   & -4 & -4 & 4 & 0 & 0 & 0 & 0 \\
 D_{37} & 12 & -12 & 4 & -4 & 0 & 0 & 0 & -4 & 0 & 4 & -4
   & 4 & 4 & -4 & 0 & 0 & 0 & 0\\
   \hline
\end{array}
\end{equation}}
{\scriptsize
\begin{equation}\label{2charto1536}
    \begin{array}{|l|lllllllllllllllllll|}
 \hline
 0 & C_{19} & C_{20} & C_{21} & C_{22} & C_{23} & C_{24}
   & C_{25} & C_{26} & C_{27} & C_{28} & C_{29} & C_{30}
   & C_{31} & C_{32} & C_{33} & C_{34} & C_{35} & C_{36}
   & C_{37} \\
 \hline
 D_1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1
   & 1 & 1 & 1 & 1 & 1 & 1 \\
 D_2 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 &
   -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 \\
 D_3 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1
   & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\
 D_4 & 1 & 1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 &
   -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 \\
 D_5 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
   & 0 & 0 & -1 & -1 & -1 & -1 \\
 D_6 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
   & 0 & 0 & -1 & 1 & -1 & 1 \\
 D_7 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 &
   -1 & -1 & 1 & 1 & 0 & 0 & 0 & 0 \\
 D_8 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1
   & 1 & -1 & -1 & 0 & 0 & 0 & 0 \\
 D_9 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 &
   1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 \\
 D_{10} & -1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & -1 & 1
   & -1 & 1 & -1 & 1 & 0 & 0 & 0 & 0 \\
 D_{11} & -1 & -1 & -1 & 1 & 1 & 1 & -1 & 1 & -1 & 1 & -1
   & 1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 \\
 D_{12} & -1 & 1 & 1 & -1 & -1 & -1 & 1 & -1 & 1 & -1 & 1
   & -1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 \\
 D_{13} & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1
   & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
 D_{14} & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 &
   -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 \\
 D_{15} & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 &
   -1 & -1 & -1 & 1 & 1 & 0 & 0 & 0 & 0 \\
 D_{16} & -1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1
   & 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 \\
 D_{17} & -1 & -1 & -1 & 1 & 1 & 1 & -1 & 1 & -1 & -1 & 1
   & -1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 \\
 D_{18} & -1 & 1 & 1 & -1 & -1 & -1 & 1 & -1 & 1 & 1 & -1
   & 1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 \\
 D_{19} & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
   0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{20} & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
   0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{21} & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 2 & 0 & -2 & 0 & 2
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{22} & 0 & 0 & 0 & 0 & 0 & 2 & 0 & -2 & 0 & 2 & 0 & -2
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{23} & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 2 & 0 & -2 & 0
   & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{24} & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & -2 & 0 & 2 & 0
   & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{25} & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 2 & 0 & 2 & 0 & -2
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{26} & 0 & 0 & 0 & 0 & 0 & 2 & 0 & -2 & 0 & -2 & 0 & 2
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{27} & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 2 & 0 & 2 & 0
   & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{28} & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & -2 & 0 & -2 & 0
   & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{29} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
   0 & 0 & 0 & 2 & 0 & -2 & 0 \\
 D_{30} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
   0 & 0 & 0 & -1 & -\sqrt{3} & 1 & \sqrt{3} \\
 D_{31} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
   0 & 0 & 0 & -1 & \sqrt{3} & 1 & -\sqrt{3} \\
 D_{32} & 0 & -2 & 2 & -2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{33} & 0 & 2 & -2 & 2 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{34} & 0 & 2 & -2 & -2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{35} & 0 & -2 & 2 & 2 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0
   & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{36} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
   0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 D_{37} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
   0 & 0 & 0 & 0 & 0 & 0 & 0\\
 \hline
\end{array}
\end{equation}
}
\par
The irreducible representations of the universal classifying group
are a fundamental tool in the classification of Arnold-Beltrami
vector fields. Indeed by choosing the various  point group  orbits
of momentum vectors in the cubic lattice, according to their
classification presented in the next section \ref{triplettoni}, and
constructing the corresponding Arnold-Beltrami fields one obtains
all of the $37$ irreducible representations of
$\mathrm{\mathrm{G_{1536}}}$. Each representation appears at least
once and some of them appear several times. Considering next the
possible subgroups $\mathcal{H}_i$ of $\mathrm{\mathrm{G_{1536}}}$
and the branching rules of  $\mathrm{\mathrm{G_{1536}}}$ irreps with
respect to $H_i$ one obtains an explicit algorithm to construct
Arnold-Beltrami vector fields with prescribed invariance space
groups $H_i$. It suffices to select the identity representation of
the subgroup in the branching rules. \textit{These are the hidden
symmetries of the Beltrami flows}. As we have discussed in the
introduction these hidden symmetries extend to the exact Navier
Stokes time dependent solutions.
\section{Classification of the 48 sublattices of the momentum
lattice and the irreps of $\mathrm{G_{1536}}$} \label{triplettoni}
Let us now analyze the action of the octahedral  group on the cubic
lattice. We define the orbits as the sets of vectors $
{\mathbf{k}}\, \in \, \Lambda$ that can be mapped one into the other
by the action of some element of the point group , namely of
$\mathrm{O_{24}}$:
\begin{equation}\label{orbitadefi}
     {\mathbf{k}}_1 \, \in \, \mathcal{O} \quad \mbox{and}
    \quad  {\mathbf{k}}_2 \, \in \, \mathcal{O} \quad \Rightarrow
    \quad \exists \, \gamma \, \in \, \mathrm{O_{24}} \; / \; \gamma\,\cdot \,
     {\mathbf{k}}_1 \, = \,  {\mathbf{k}}_2
\end{equation}
In the case of the cubic lattice there are four type of orbits
\subsection{Orbits of length 6} Each of these orbits is of the
following form:
\begin{equation}\label{orbita6}
\mathcal{O}_6 \, = \,     \left\{
\begin{array}{llllll}
 \{0,0,-n\}, & \{0,0,n\}, & \{0,-n,0\}, & \{0,n,0\}, &
   \{-n,0,0\}, & \{n,0,0\}
\end{array}
\right\}
\end{equation}
where $n \, \in \, \mathbb{Z}$ is any integer number. The six
vectors belonging to this orbit can be seen as the vertices of a
regular octahedron (see fig.\ref{orbit6cub})
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{octa6orbit.png}
\end{center}
\caption{\label{orbit6cub}{\it  The length 6 orbit of the octahedral
group acting on the cubic lattice corresponds to the lattice points
marked in red that can be viewed as the six vertices of a regular
octahedron.}}
\end{figure}
\subsection{Orbits of length 8}
Each of these orbits is of the following form
\begin{equation}\label{orbita8}
\mathcal{O}_8 \, = \,     \left\{
\begin{array}{llll}
 \{-n,-n,-n\}, & \{-n,-n,n\}, & \{-n,n,-n\}, & \{-n,n,n\}, \\
   \{n,-n,-n\}, & \{n,-n,n\}, & \{n,n,-n\}, & \{n,n,n\}\\
\end{array}
\right\}
\end{equation}
where $n \, \in \, \mathbb{Z}$ is any integer number.The 8 vectors
belonging to this orbit can be seen as the vertices of a cube (see
fig.\ref{orbit8cub})
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{octa8orbit.png}
\end{center}
\caption{\label{orbit8cub}{\it  The length 8 orbit of the octahedral
group acting on the cubic lattice corresponds to the lattice points
marked in red that can be viewed as the 8 vertices of a cube.}}
\end{figure}
\subsection{Orbits of length 12}
Each of these orbits is of the following form:
\begin{equation}\label{orbita12}
\mathcal{O}_{12} \, = \,     \left\{
\begin{array}{llll}
 \{0,-n,-n\}, & \{0,-n,n\}, & \{0,n,-n\}, & \{0,n,n\}, \\
   \{-n,0,-n\}, & \{-n,0,n\}, & \{-n,-n,0\}, & \{-n,n,0\}, \\
   \{n,0,-n\}, & \{n,0,n\}, & \{n,-n,0\}, & \{n,n,0\}
\end{array}
\right\}
\end{equation}
where $n \, \in \, \mathbb{Z}$ is any integer number. The 12 vectors
belonging to this orbit can be seen as the middle points  of the
edges of a cube (see fig.\ref{orbit12cub})
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{octa12orbit.png}
\end{center}
\caption{\label{orbit12cub}{\it  The length 12 orbit of the
octahedral group acting on the cubic lattice corresponds to the
lattice points marked in red that can be viewed as the middle points
of the edges of a cube.}}
\end{figure}
\subsection{Orbits of length 24}
Each of these orbits is of the following form:
\begin{equation}\label{orbita24}
\mathcal{O}_{24} \, = \,     \left\{
\begin{array}{llll}
 \{-p,-q,r\}, & \{-p,q,-r\}, & \{-p,-r,-q\}, & \{-p,r,q\}, \\
   \{p,-q,-r\}, & \{p,q,r\}, & \{p,-r,q\}, & \{p,r,-q\}, \\
   \{-q,-p,-r\}, & \{-q,p,r\}, & \{-q,-r,p\}, & \{-q,r,-p\},\\
   \{q,-p,r\}, & \{q,p,-r\}, & \{q,-r,-p\}, & \{q,r,p\}, \\
   \{-r,-p,q\}, & \{-r,p,-q\}, & \{-r,-q,-p\}, & \{-r,q,p\},\\
    \{r,-p,-q\}, & \{r,p,q\}, & \{r,-q,p\}, & \{r,q,-p\},\\
\end{array}
\right\}
\end{equation}
where $\{p, \, q, \, r \}\, \in \, \mathbb{Z}$ is any triplet of
integer numbers that are not all three equal in absolute value.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{octa24orbit.png}
\end{center}
\caption{\label{orbit24cub}{\it  A possible length 24 orbit of the
octahedral group acting on the cubic lattice corresponds to the
lattice points marked in black. In this case the orbit is generated
by the vector $\{1,2,3\}$ and all the orbit points belong to the
intersection of the cubic lattice $\Lambda_{cubic}$ with a sphere of
radius $r=\sqrt{14}$.}}
\end{figure}
\par
\subsection{Classification of the $48$ types of orbits}
The first observation is that the  group
$\mathrm{\mathrm{G_{1536}}}$ has a finite number of irreducible
representations so that the number of different types of
Arnold-Beltrami vector fields has also got to be finite,  namely as
many as the 37 irreps, times the number of different ways to obtain
them from orbits of length 6,8,12 or 24. The second observation is
the key role of the number $4$ introduced by Frobenius congruences
which was already the clue to the definition of
$\mathrm{\mathrm{G_{1536}}}$. What we should expect is that the
various orbits should be defined with integers modulo $4$ in other
words that we should just consider the possible octahedral orbits on
a lattice with coefficients in $\mathbb{Z}_4$ rather than
$\mathbb{Z}$. The easy guess, which is confirmed by computer
calculations, is that the pattern of $\mathrm{\mathrm{G_{1536}}}$
representations obtained from the construction of Arnold-Beltrami
vector fields according to the algorithm of section \ref{algoritmo}
depends only on the equivalence classes of momentum orbits modulo
$4$. Hence we have a finite number of such orbits and a finite
number of Arnold-Beltrami vector fields which we presently describe.
Let us stress that an embryo of the exhaustive classification of
orbits we are going to present was introduced by Arnold in his paper
\cite{arnoldorussopapero}. Arnold's one  was only an embryo of the
complete classification for the following two reasons:
\begin{enumerate}
\item The type of momenta orbits were partitioned  according to $odd$ and $even$
(namely according to $\mathbb{Z}_2$, rather than $\mathbb{Z}_4$)
\item The classifying group was taken to be the crystallographic $\mathrm{GS_{24}}$,
as defined  in the appendices of \cite{Fre:2015mla} and already
discussed in eq.(\ref{TScap}) and following lines.
\end{enumerate}
Let us then present the complete classification of point orbits in
the momentum lattice. First we subdivide the momenta into five
groups:
\begin{description}
  \item[A)] Momenta of type $\{a,0,0\}$ which generate $\mathrm{O_{24}}$
  orbits of length 6 and representations of the universal
  group $\mathrm{G}_{1536}$ also of dimensions 6.
  \item[B)]Momenta of type $\{a,a,a\}$ which generate $\mathrm{O_{24}}$ orbits
  of length 8 and representations of the universal group $\mathrm{G}_{1536}$
  also of dimensions 8.
  \item[C)] Momenta of type $\{a,a,0\}$ which generate $\mathrm{O_{24}}$
  orbits of length 12 and representations of the universal
  group $\mathrm{G}_{1536}$ also of dimensions 12.
  \item[D)] Momenta of type $\{a,a,b\}$ which generate $\mathrm{O_{24}}$
  orbits of length 24 and representations
  of the universal group $\mathrm{G}_{1536}$ also of dimensions 24.
  \item[E)] Momenta of type $\{a,b,c\}$ which
  generate $\mathrm{O_{24}}$ orbits of length 24 and
  representations of the universal group $\mathrm{G}_{1536}$ of dimensions 48.
\end{description}
The reason why in the cases A)\dots D) the dimension of the
representation $\mathfrak{R}\left(\mathrm{\mathrm{G_{1536}}}\right)$
coincides with the dimension $|\mathcal{O}|$ of the orbit  is
simple. For each momentum in the orbit ($\forall\mathbf{k}_i\, \in
\, \mathcal{O}$) also its negative appears in the same orbit ($- \,
\mathbf{k}_i\, \in \, \mathcal{O}$), hence the number of arguments
$\Theta_i \, \equiv \, 2\pi \,\mathbf{k}_i \cdot \mathbf{x}$ of the
independent trigonometric functions $\sin \left( \Theta_i\right)$
and $\cos \left(\Theta_i\right)$ is $\ft 12 \times 2 |\mathcal{O}|
\, = \, |\mathcal{O}|$ since $\sin \left( \pm\Theta_i\right) \, = \,
\pm  \sin \left( \Theta_i\right)$ and $\cos \left(
\pm\Theta_i\right) \, = \, \cos \left( \Theta_i\right)$.
\par In
case E), instead, the negatives of all the members of the orbit
$\mathcal{O}$ are not in $\mathcal{O}$. The number of independent
trigonometric functions is therefore 48 and such is the dimension of
the representation
$\mathfrak{R}\left(\mathrm{\mathrm{G_{1536}}}\right)$.
\par
In each of the five groups one still has  to reduce the entries to
$\mathbb{Z}_4$, namely to consider their equivalence class
$\mathrm{mod}\,4$. Each different choice of the pattern of
$\mathbb{Z}_4$ classes appearing in an orbit leads to a different
decomposition of the representation into irreducible representation
of $\mathrm{G}_{1536}$. A simple consideration of the combinatorics
leads to the conclusion that there are in total $48$ cases to be
considered. The very significant result is that all of the $37$
irreducible representations of $\mathrm{G}_{1536}$ appear at least
once in the list of these decompositions. Hence for all the
\textit{irreps} of this group one can find a corresponding Beltrami
field and for some \textit{irreps} such a Beltrami field admits a
few inequivalent realizations. The list of the $48$ distinct types
of momenta is the following one:
\begin{enumerate}
\item $\mathbf{k} \, = \,  \{0,0, 1+4 \rho \}$
\item $\mathbf{k} \, = \,  \{0,0, 2+4 \rho \}$
\item $\mathbf{k} \, = \,  \{0,0, 3+4 \rho \}$
\item $\mathbf{k} \, = \,  \{0,0, 4+4 \rho \}$
\item $\mathbf{k} \, = \,  \left\{1+4\mu ,1+4 \mu ,1+4 \mu \right\}$
\item $\mathbf{k} \, = \,  \left\{2+4 \mu ,2+4 \mu ,2+4 \mu \right\}$
\item $\mathbf{k} \, = \,  \left\{3+4 \mu ,3+4 \mu ,3+4 \mu \right\}$
\item $\mathbf{k} \, = \,  \left\{4+4 \mu ,4+4 \mu ,4+4 \mu \right\}$
\item $\mathbf{k} \, = \,  \left\{0 ,1+4 \nu ,1+4 \nu \right\}$
\item $\mathbf{k} \, = \,  \left\{0 ,2+4 \nu ,2+4 \nu \right\}$
\item $\mathbf{k} \, = \,  \left\{0 ,3+4 \nu ,3+4 \nu \right\}$
\item $\mathbf{k} \, = \,  \left\{0 ,4+4 \nu ,4+4 \nu \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4  \mu ,1+4 \mu ,2+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4 \mu ,1+4 \mu ,3+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4 \mu ,1+4 \mu ,4+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4\mu ,1+4 \mu ,5+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \{1+4 \mu ,2+4 \mu ,2+4 \rho \}$
\item $\mathbf{k} \, = \,  \left\{2+4\mu ,2+4 \mu ,6+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{2+4  \mu ,2+4 \mu ,3+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{2+4\mu ,2+4 \mu ,4+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4 \mu ,3+4 \mu ,3+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{2+4 \mu ,3+4 \mu ,3+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{3+4  \mu ,3+4 \mu ,7+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4  \mu ,4+4 \mu ,4+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \{2+4 \mu ,4+4 \mu ,4+4 \rho \}$
\item $\mathbf{k} \, = \,  \left\{3+4  \mu ,4+4 \mu ,4+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{4+4 \mu ,4+4 \mu ,8+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{3+4  \mu ,3+4 \mu ,4+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{4+4 \mu ,8+4 \nu ,12+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \{1+4 \mu ,4+4 \nu ,8+4 \rho \}$
\item $\mathbf{k} \, = \,  \left\{2+4 \mu ,4+4 \nu ,8+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{3+4  \mu ,4+4 \nu ,8+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4 \mu ,2+4 \nu ,4+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4 \mu ,3+4 \nu ,4+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{2+4 \mu ,4+4 \nu ,6+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{2+4  \mu ,3+4 \nu ,4+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4 \mu ,5+4 \nu ,9+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4  \mu ,2+4 \nu ,5+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4  \mu ,3+4 \nu ,5+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4  \mu ,2+4 \nu ,6+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4 \mu ,2+4 \nu ,3+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4 \mu ,3+4 \nu ,7+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{2+4 \mu ,6+4 \nu ,10+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \{2+4 \mu ,3+4 \nu ,6+4 \rho \}$
\item $\mathbf{k} \, = \,  \left\{2+4 \mu ,3+4 \nu ,7+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{3+4  \mu ,7+4 \nu ,11+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{1+4 \mu ,4+4 \nu ,5+4 \rho \right\}$
\item $\mathbf{k} \, = \,  \left\{3+4  \mu ,4+4 \nu ,7+4 \rho \right\}$
\end{enumerate}
where $\mu,\nu,\rho \, \in \, \mathbb{Z}$. The simplest and lowest
lying representative of each of the $48$ classes of equivalent
momenta is obtained choosing $\mu \, = \, \nu \, = \, 0$.
\subsection{The 48 orbits type and the
irreps of the Universal Classifying Group} In this subsection,
quoting the results obtained in \cite{Fre:2015mla} for each of the
$48$ classes enumerated above we provide the decomposition of the
corresponding Beltrami vector field parameter space into
$\mathrm{\mathrm{G_{1536}}}$ irreducible representations. These
results are the outcome of extensive MATHEMATICA calculations that
were performed with purposely written codes. As already stressed the
most relevant point is that all the $37$ irreps of the Classifying
Group are reproduced: this is the main reason for its name.
\subsubsection{Classes
of momentum vectors yielding orbits of length 6: $\{$a,0,0$\}$}
\label{mortadella6} \noindent\(
\pmb{}\\
\pmb{\text{ Class of the momentum vector  = } \{0,0, 1+4 \rho \}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 6}\\
\pmb{\text{ Orbit   = } D_{23}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \{0,0, 2+4 \rho \}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 6}\\
\pmb{\text{ Orbit   = } D_{19}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \{0,0, 3+4 \rho \}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 6}\\
\pmb{\text{ Orbit   = } D_{24}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \{0,0, 4+4 \rho \}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 6}\\
\pmb{\text{ Orbit   = } D_7[\mathrm{G}_{1536},3]+D_8[\mathrm{G}_{1536},3]}\\
\pmb{ }\)
\subsubsection{Classes of momentum vectors yielding orbits
of length 8: $\{$a,a,a$\}$}
\noindent\(
\pmb{}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4\mu ,1+4 \mu ,1+4 \mu \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 8}\\
\pmb{\text{ Orbit   = } D_{30}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4 \mu ,2+4 \mu ,2+4 \mu \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 8}\\
\pmb{\text{ Orbit   = } D_6[\mathrm{G}_{1536},2]+D_{17}[\mathrm{G}_{1536},3]+D_{18}[\mathrm{G}_{1536},3]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{3+4 \mu ,3+4 \mu ,3+4 \mu \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 8}\\
\pmb{\text{ Orbit   = } D_{31}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{4+4 \mu ,4+4 \mu ,4+4 \mu \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 8}\\
\pmb{\text{ Orbit   = } D_5[\mathrm{G}_{1536},2]+D_7[\mathrm{G}_{1536},3]+D_8[\mathrm{G}_{1536},3]}\\
\pmb{ }\)
\subsubsection{Classes of momentum vectors yielding orbits
of length 12: $\{$0,a,a$\}$} \noindent\(
\pmb{}\\
\pmb{\text{ Class of the momentum vector  = } \left\{0 ,1+4 \nu ,1+4 \nu \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 12}\\
\pmb{\text{ Orbit   = } D_{32}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{0 ,2+4 \nu ,2+4 \nu \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 12}\\
\pmb{\text{ Orbit   = } D_{13}[\mathrm{G}_{1536},3]+D_{15}[\mathrm{G}_{1536},3]+D_{20}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{0 ,3+4 \nu ,3+4 \nu \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 12}\\
\pmb{\text{ Orbit   = } D_{32}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{0 ,4+4 \nu ,4+4 \nu \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 12}\\
\pmb{\text{ Orbit   = }
D_2[\mathrm{G}_{1536},1]+D_5[\mathrm{G}_{1536},2]+D_7[\mathrm{G}_{1536},3]+2
D_8[\mathrm{G}_{1536},3]}\\
\pmb{ }\)
\subsubsection{Classes of momentum vectors yielding orbits
of length 24: $\{$a,a,b$\}$} \noindent\(
\pmb{}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4  \mu ,1+4 \mu ,2+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{34}[\mathrm{G}_{1536},12]+D_{35}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4 \mu ,1+4 \mu ,3+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{29}[\mathrm{G}_{1536},8]+D_{30}[\mathrm{G}_{1536},8]+D_{31}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4 \mu ,1+4 \mu ,4+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{32}[\mathrm{G}_{1536},12]+D_{33}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4\mu ,1+4 \mu ,5+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{29}[\mathrm{G}_{1536},8]+D_{30}[\mathrm{G}_{1536},8]+D_{31}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \{1+4 \mu ,2+4 \mu ,2+4 \rho \}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{25}[\mathrm{G}_{1536},6]+D_{26}[\mathrm{G}_{1536},6]+D_{27}[\mathrm{G}_{1536},6]+D_{28}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4\mu ,2+4 \mu ,6+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = }
D_3[\mathrm{G}_{1536},1]+D_4[\mathrm{G}_{1536},1]+2
D_6[\mathrm{G}_{1536},2]+3
D_{17}[\mathrm{G}_{1536},3]+3 D_{18}[\mathrm{G}_{1536},3]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4  \mu ,2+4 \mu ,3+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{25}[\mathrm{G}_{1536},6]+D_{26}[\mathrm{G}_{1536},6]+D_{27}[\mathrm{G}_{1536},6]+D_{28}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4\mu ,2+4 \mu ,4+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = }
D_{13}[\mathrm{G}_{1536},3]+D_{14}[\mathrm{G}_{1536},3]+D_{15}[\mathrm{G}_{1536},3]+D_{16}[\mathrm{G}_{1536},3]+2
D_{20}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4 \mu ,3+4 \mu ,3+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{29}[\mathrm{G}_{1536},8]+D_{30}[\mathrm{G}_{1536},8]+D_{31}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4 \mu ,3+4 \mu ,3+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{34}[\mathrm{G}_{1536},12]+D_{35}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{3+4  \mu ,3+4 \mu ,7+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{29}[\mathrm{G}_{1536},8]+D_{30}[\mathrm{G}_{1536},8]+D_{31}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4  \mu ,4+4 \mu ,4+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{21}[\mathrm{G}_{1536},6]+D_{22}[\mathrm{G}_{1536},6]+D_{23}[\mathrm{G}_{1536},6]+D_{24}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \{2+4 \mu ,4+4 \mu ,4+4 \rho \}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = }
D_9[\mathrm{G}_{1536},3]+D_{10}[\mathrm{G}_{1536},3]+D_{11}[\mathrm{G}_{1536},3]+D_{12}[\mathrm{G}_{1536},3]+2
D_{19}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{3+4  \mu ,4+4 \mu ,4+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{21}[\mathrm{G}_{1536},6]+D_{22}[\mathrm{G}_{1536},6]+D_{23}[\mathrm{G}_{1536},6]+D_{24}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{4+4 \mu ,4+4 \mu ,8+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = }
D_1[\mathrm{G}_{1536},1]+D_2[\mathrm{G}_{1536},1]+2
D_5[\mathrm{G}_{1536},2]+3
D_7[\mathrm{G}_{1536},3]+3 D_8[\mathrm{G}_{1536},3]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{3+4  \mu ,3+4 \mu ,4+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 24}\\
\pmb{\text{ Orbit   = } D_{32}[\mathrm{G}_{1536},12]+D_{33}[\mathrm{G}_{1536},12]}\\
\pmb{ }\)
\subsubsection{Classes of momentum vectors yielding point
orbits of length 24 and $\mathrm{G_{1536}}$ representations of
dimensions 48: $\{$a,b,c$\}$} \noindent\(
\pmb{}\\
\pmb{\text{ Class of the momentum vector  = } \left\{4+4 \mu ,8+4 \nu ,12+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_1[\mathrm{G}_{1536},1]+2
D_2[\mathrm{G}_{1536},1]+4 D_5[\mathrm{G}_{1536},2]+6
D_7[\mathrm{G}_{1536},3]+6 D_8[\mathrm{G}_{1536},3]}\\
\pmb{\text{ Class of the momentum vector  = } \{1+4 \mu ,4+4 \nu ,8+4 \rho \}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{21}[\mathrm{G}_{1536},6]+2
D_{22}[\mathrm{G}_{1536},6]+2
D_{23}[\mathrm{G}_{1536},6]+2 D_{24}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4 \mu ,4+4 \nu ,8+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_9[\mathrm{G}_{1536},3]+2
D_{10}[\mathrm{G}_{1536},3]+2 D_{11}[\mathrm{G}_{1536},3]+2
D_{12}[\mathrm{G}_{1536},3]+4 D_{19}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{3+4  \mu ,4+4 \nu ,8+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{21}[\mathrm{G}_{1536},6]+2
D_{22}[\mathrm{G}_{1536},6]+2
D_{23}[\mathrm{G}_{1536},6]+2 D_{24}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4 \mu ,2+4 \nu ,4+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{36}[\mathrm{G}_{1536},12]+2 D_{37}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4 \mu ,3+4 \nu ,4+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{32}[\mathrm{G}_{1536},12]+2 D_{33}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4 \mu ,4+4 \nu ,6+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{13}[\mathrm{G}_{1536},3]+2
D_{14}[\mathrm{G}_{1536},3]+2
D_{15}[\mathrm{G}_{1536},3]+2 D_{16}[\mathrm{G}_{1536},3]+4 D_{20}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4  \mu ,3+4 \nu ,4+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{36}[\mathrm{G}_{1536},12]+2 D_{37}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4 \mu ,5+4 \nu ,9+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{29}[\mathrm{G}_{1536},8]+2
D_{30}[\mathrm{G}_{1536},8]+2
D_{31}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4  \mu ,2+4 \nu ,5+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{34}[\mathrm{G}_{1536},12]+2 D_{35}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4  \mu ,3+4 \nu ,5+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{29}[\mathrm{G}_{1536},8]+2
D_{30}[\mathrm{G}_{1536},8]+2
D_{31}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4  \mu ,2+4 \nu ,6+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{25}[\mathrm{G}_{1536},6]+2
D_{26}[\mathrm{G}_{1536},6]+2
D_{27}[\mathrm{G}_{1536},6]+2 D_{28}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4 \mu ,2+4 \nu ,3+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{34}[\mathrm{G}_{1536},12]+2 D_{35}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4 \mu ,3+4 \nu ,7+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{29}[\mathrm{G}_{1536},8]+2
D_{30}[\mathrm{G}_{1536},8]+2
D_{31}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4 \mu ,6+4 \nu ,10+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_3[\mathrm{G}_{1536},1]+2
D_4[\mathrm{G}_{1536},1]+4 D_6[\mathrm{G}_{1536},2]+6
D_{17}[\mathrm{G}_{1536},3]+6 D_{18}[\mathrm{G}_{1536},3]}\\
\pmb{\text{ Class of the momentum vector  = } \{2+4 \mu ,3+4 \nu ,6+4 \rho \}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{25}[\mathrm{G}_{1536},6]+2
D_{26}[\mathrm{G}_{1536},6]+2
D_{27}[\mathrm{G}_{1536},6]+2 D_{28}[\mathrm{G}_{1536},6]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{2+4 \mu ,3+4 \nu ,7+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{34}[\mathrm{G}_{1536},12]+2 D_{35}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{3+4  \mu ,7+4 \nu ,11+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{29}[\mathrm{G}_{1536},8]+2
D_{30}[\mathrm{G}_{1536},8]+2
D_{31}[\mathrm{G}_{1536},8]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{1+4 \mu ,4+4 \nu ,5+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{32}[\mathrm{G}_{1536},12]+2 D_{33}[\mathrm{G}_{1536},12]}\\
\pmb{\text{ Class of the momentum vector  = } \left\{3+4  \mu ,4+4 \nu ,7+4 \rho \right\}}\\
\pmb{\text{ Dimension of the $\mathrm{G}_{1536}$  representation  = } 48}\\
\pmb{\text{ Orbit   = } 2 D_{32}[\mathrm{G}_{1536},12]+2 D_{33}[\mathrm{G}_{1536},12]}\\
\pmb{ }\)
\subsubsection{The interpretation of the 48 momentum
classes as sublattices of the cubic lattice}  The union of the
orbits of the 48 vector classes for all values of the integer
parameters $\mu,\nu,\rho$ constitute infinite sublattices of the
momentum lattice.
\par
Given the class
$\mathbf{k}^{p,q,r}=\left\{p+4\mu,q+4\nu,r+4\rho\right\}$ and the
corresponding orbit of each vector in the class
$\mathcal{O}^{(p,q,r)}\left(\mu,\nu,\rho\right)$ considering all
$\mu,\nu,\rho \in \mathbb{Z}$ we obtain a sublattice of the original
lattice:
\begin{equation}\label{sublattice}
    \Lambda^{p,q,r} \, \equiv \, \bigoplus_{\mu,\nu,\rho}^\infty
    \,\mathcal{O}^{(p,q,r)} \, \subset \Lambda_{cubic}
\end{equation}
Most of these sublattices are $1$ dimensional
$\mathbb{Z}\hookrightarrow \mathbb{Z}\times \mathbb{Z}\times
\mathbb{Z}$ some are two dimensional $\mathbb{Z}\times
\mathbb{Z}\hookrightarrow \mathbb{Z}\times \mathbb{Z}\times
\mathbb{Z}$  and some are three dimensional $\mathbb{Z}\times
\mathbb{Z}\times \mathbb{Z} \hookrightarrow \mathbb{Z}\times
\mathbb{Z}\times \mathbb{Z}$.
\par
For instance the sublattice $\Lambda^{0,0,1}$ is formed by all the
six dimensional orbits of the vectors of type $\{0,0,1+4\mu\}$ with
$\mu\in \mathbb{Z}$. A picture of the immersion of the points of
this sublattice in the full cubic lattice is provided in
fig.\ref{ceronetto1}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{sublat6.png}
\end{center}
\caption{\label{ceronetto1}{\it   A picture of the immersion of the
sublattice $\Lambda^{0,0,1}$ into the full cubic lattice. The points
of $\Lambda^{0,0,1}$ are painted in red on the background of the
grid of the cubic lattice.}}
\end{figure}
Similarly the sublattice $\Lambda^{1,1,1}$ is formed by all the
eight dimensional orbits of the vectors of type
$\{1+4\mu,1+4\mu,1+4\mu\}$ with $\mu\in \mathbb{Z}$. A picture of
the immersion of the points of this sublattice in the full cubic
lattice is provided in fig.\ref{ceronetto2}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{sublat8.png}
\end{center}
\caption{\label{ceronetto2}{\it   A picture of the immersion of the
sublattice $\Lambda^{1,1,1}$ into the full cubic lattice. The points
of $\Lambda^{1,1,1}$ are painted in red on the background of the
grid of the cubic lattice.}}
\end{figure}
Finally we display sublattice $\Lambda^{1,1,2}$ which is formed by
all the eight dimensional orbits of the vectors of type
$\{1+4\mu,1+4\mu,2+4\nu\}$ with $\mu,\nu\in \mathbb{Z}$. A picture
of the immersion of the points of this sublattice in the full cubic
lattice is provided in fig.\ref{ceronetto3}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{sublat24.png}
\end{center}
\caption{\label{ceronetto3}{\it   A picture of the immersion of the
sublattice $\Lambda^{1,1,2}$ into the full cubic lattice. The points
of $\Lambda^{1,1,2}$ are painted in red on the background of the
grid of the cubic lattice.}}
\end{figure}
\section{The universal classifying group $\mathfrak{U}_{72}$ for the Hexagonal Lattice $\Lambda _{Hex}$}
In this section following  the same procedure utilized in the cubic
case, namely Frobenius congruences, we identify the Universal
Classifying Group for the hexagonal lattice with point group
$\mathfrak{P}_{\Lambda_{Hex}}\, = \, \mathrm{Dih_{6}}$ and we
discover that it is a group with $72$ elements.
\subsection{Frobenius congruences for $\mathrm{Dih_6}$}
Utilizing the block triangular representation for the semidirect
product we introduce the two candidate generators
$\hat{\mathcal{A}}$ and $\hat{\mathcal{B}}$ as it follows:
\begin{equation}\label{laperonista}
    \hat{\mathcal{A}}\, = \,\left(
\begin{array}{ccc|c}
 \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 & \alpha _1 \\
 -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 & \alpha _2 \\
 0 & 0 & 1 & \alpha _3 \\
 \hline
 0 & 0 & 0 & 1 \\
\end{array}
\right) \quad ; \quad \hat{\mathcal{B}}\, = \,\left(
\begin{array}{ccc|c}
 -1 & 0 & 0 & \beta _1 \\
 0 & 1 & 0 & \beta _2 \\
 0 & 0 & -1 & \beta _3 \\
 \hline
 0 & 0 & 0 & 1 \\
\end{array}
\right)
\end{equation}
and we impose the three conditions:
\begin{equation}\label{solstizio}
    \hat{\mathcal{A}}^6 \, \in \, \hat{\Lambda} \quad ; \quad \hat{\mathcal{B}}^2 \, \in \,\hat{\Lambda}\quad ; \quad
  \left( \hat{\mathcal{B}}\cdot\hat{\mathcal{A}} \right)^2 \, \in \,  \hat{\Lambda}
\end{equation}
where the lattice subgroup is embedded in the inhomogeneous point
group $\mathfrak{Ip}_{\Lambda_{Hex}}$ as it is specified below:
\begin{equation}\label{cardenson}
    \mathfrak{Ip}_{\Lambda_{Hex}} \, \supset \, \hat{\Lambda} \, =
    \,\left\{ \left(
                \begin{array}{ccc|c}
                  1 & 0 & 0 & \sqrt{2} m_1-\frac{m_2}{\sqrt{2}} \\
                  0 & 1 & 0 & \sqrt{\frac{3}{2}} m_2 \\
                  0 & 0 & 1 & \sqrt{\frac{3}{2}} m_3 \\
                  \hline
                  0 & 0 & 0 & 0 \\
                \end{array}
              \right) \, \parallel \, m_{1,2,3} \, \in \,
              \mathbb{Z}\right \}
\end{equation}
By explicit calculation we find:
\begin{eqnarray}
\label{esplicitone}
   \hat{\mathcal{A}}^6 &=&  \left(
\begin{array}{cccc}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 6 \alpha _3 \\
 0 & 0 & 0 & 1 \\
\end{array}
\right) \quad ; \quad \hat{\mathcal{B}}^2 \, = \, \left(
\begin{array}{ccc|c}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 2 \beta _2 \\
 0 & 0 & 1 & 0 \\
 \hline
 0 & 0 & 0 & 1 \\
\end{array}
\right)\nonumber \\
  \left( \hat{\mathcal{B}}\cdot\hat{\mathcal{A}} \right)^2 &=& \left(
\begin{array}{ccc|c}
 1 & 0 & 0 & \frac{1}{2} \left(-\alpha _1-\sqrt{3} \alpha _2+\beta _1-\sqrt{3} \beta
   _2\right) \\
 0 & 1 & 0 & \frac{1}{2} \left(\sqrt{3} \alpha _1+3 \alpha _2-\sqrt{3} \beta _1+3
   \beta _2\right) \\
 0 & 0 & 1 & 0 \\
 \hline
 0 & 0 & 0 & 1 \\
\end{array}
\right)
\end{eqnarray}
Next introducing the generic translation group element
\begin{equation}\label{genericT}
    T \, = \, \left(
\begin{array}{ccc|c}
 1 & 0 & 0 & \text{t}_{1} \\
 0 & 1 & 0 & \text{t}_{2} \\
 0 & 0 & 1 & \text{t}_{3} \\
 \hline
 0 & 0 & 0 & 1 \\
\end{array}
\right) \quad ; \quad \text{t}_{1,2,3} \, \in \, \mathbb{R}
\end{equation}
by conjugating the two generators
$\hat{\mathcal{A}},\hat{\mathcal{B}}$ we find:
\begin{eqnarray}\label{sposalizio}
    T^{-1}\,\hat{\mathcal{A}}\, T & = & \left(
\begin{array}{ccc|c}
 \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \left(2 \alpha
   _1+\text{t}_{1}-\sqrt{3} \text{t}_{2}\right) \\
 -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 & \frac{1}{2} \left(2 \alpha _2+\sqrt{3}
   \text{t}_{1}+\text{t}_{2}\right) \\
 0 & 0 & 1 & \alpha_3 \\
 \hline
 0 & 0 & 0 & 1 \\
\end{array}
\right) \nonumber\\
T^{-1}\,\hat{\mathcal{B}}\, T & = &\left(
\begin{array}{ccc|c}
 -1 & 0 & 0 & \beta _1+2 \text{t}_{1} \\
 0 & 1 & 0 & \beta _2\\
 0 & 0 & -1 & \beta _3+2 \text{t}_{3} \\
 \hline
 0 & 0 & 0 & 1 \\
\end{array}
\right)
\end{eqnarray}
Hence the parameters $\mathrm{t}_{1,2}$ can be used to set
$\alpha_{1,2} \, = \, 0$, while the parameter $\mathrm{t}_3$ can be
utilized to set $\beta_3 \, = 0$. Inserting such a gauge chocie in
the conditions (\ref{solstizio}), in view of eq.(\ref{esplicitone})
and (\ref{cardenson}) we finally get:
\begin{equation}\label{argentina}
    \hat{\mathcal{A}}\, = \,\left(
\begin{array}{ccc|c}
 \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 & 0 \\
 -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 & 0 \\
 0 & 0 & 1 & \frac{1}{2 \sqrt{6}} \\
 \hline
 0 & 0 & 0 & 1 \\
\end{array}
\right) \quad ; \quad \hat{\mathcal{B}}\, = \,\left(
\begin{array}{ccc|c}
 -1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & -1 & 0 \\
 \hline
 0 & 0 & 0 & 1 \\
\end{array}
\right)
\end{equation}
Following the same logic utilized in the cubic case the result that
we obtain from eq.(\ref{argentina}) is that the only fractional
translations to be considered are in the $z$-direction and that they
are of length $1/6$ of the lattice spacing. Indeed the column vector
appearing in the $\hat{\mathcal{A}}$-generator, \textit{i.e.}
\begin{equation}\label{corinzio}
    \left \{ 0,0,\frac{1}{2 \sqrt{6}} \right \} \, = \, \frac{1}{6}
\mathbf{w}_3
\end{equation}
is the generator of translational $\mathbb{Z}_6$ subgroup and the
Universal Classifying Group for the hexagonal lattice turns out to
be:
\begin{equation}\label{U72definio}
    \mathfrak{U}_{72} \, \equiv \, \mathbb{Z}_6 \, \ltimes \,\mathrm{Dih_6}
\end{equation}

\subsection{Structure and irreps of $\mathfrak{U}_{72}$}
Utilizing the information obtained from Frobenius congruences we
know that the abstract structure of the group that we name
\(\mathfrak{U}_{72}\) is the following one:
\begin{equation}\label{structU72}
\mathfrak{U}_{72}\,
=\,\mathbb{Z}_2\underset{\text{semidirect}}{\ltimes}\left(\mathbb{Z}_6\otimes
\mathbb{Z}_6\right)
\end{equation}
The generators and relations defining this group are as follows. We
have just three generators named
$\{\mathcal{A}$,$\mathcal{B}$,$\mathcal{T}\}$ that obey the
relations:
\begin{equation}\label{generDih6Rela}
\mathcal{A}^6=\mathcal{B}^2=\mathcal{T}^6=(\mathcal{B}\mathcal{A})^2=(\mathcal{B}\mathcal{T})^2=\mathcal{E}\text{
};\text{      }\mathcal{A}\mathcal{T}=\mathcal{T}\mathcal{A}
\end{equation}
In the case of the hexagonal lattice $\mathcal{A}$,$\mathcal{B}$ are
realized as proper rotations belonging to $\mathrm{SO(3)}$ and they
generate the dihedral group $\text{Dih}_6$. The generator
$\mathcal{T}$ is a translation (modulo lattice). However if we
suppress the generator $\mathcal{A}$ we obtain another dihedral
group $\text{Dih}_6\,\subset \,\mathfrak{U}_{72}$ realized partially
by rotations, partially by translations. The group
$\mathfrak{U}_{72}$ contains a maximal normal abelian subgroup that
we name $\mathfrak{N}_{36}\simeq \mathbb{Z}_6\otimes \mathbb{Z}_6$
which is generated by $\mathcal{A}$ and $\mathcal{T}$:
\begin{equation}\label{normalosottogruppo36}
   \mathfrak{U}_{72} \, \rhd \, \mathfrak{N}_{36}
\end{equation}
This fact is fundamental in order to construct all the irreducible
representations of \(\mathfrak{U}_{72}\) with the iterative
procedure that can be applied to solvable groups (see section
\ref{startegos}).
\subsubsection{The auxiliary four dimensional representation of
$\mathfrak{U}_{72}$} \label{U72coniugato}
As we are going to see
below, none of the irreducible representation of $\mathfrak{U}_{72}$
is faithful. In order to study the algebraic structure of
$\mathfrak{U}_{72}$ and its organization in conjugacy classes, we
need a faithful representation. The smallest we found is in four
dimension.
\par
The auxiliary four dimensional representation is generated as it
follows :
\begin{eqnarray}
  \mathcal{A} &=& \left(
\begin{array}{cccc}
 \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 & 0 \\
 -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1 \\
\end{array}
\right) \quad ; \quad \mathcal{T}\, = \, \left(
\begin{array}{cccc}
 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & \frac{1}{2} & \frac{\sqrt{3}}{2} \\
 0 & 0 & -\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\end{array}
\right) \nonumber\\
  \mathcal{B} &=& \left(
\begin{array}{cccc}
 1 & 0 & 0 & 0 \\
 0 & -1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & -1 \\
\end{array}
\right)
\end{eqnarray}
From the above generators we obtain an explicit form of all the 72
elements that are organized in 24 conjugacy classes as it is
displayed in the table below:\\
\begin{doublespace}
\noindent\(\text{Class }1\mid\text{ order of elements = }1\mid\text{ $\#$ of elem in class = }1\mid\text{ representative = }\mathcal{E}\\
\text{Class }2\mid\text{ order of elements = }2\mid\text{ $\#$ of elem in class = }1\mid\text{ representative = }\mathcal{A}^3.\mathcal{T}^3\\
\text{Class }3\mid\text{ order of elements = }2\mid\text{ $\#$ of elem in class = }1\mid\text{ representative = }\mathcal{A}^3\\
\text{Class }4\mid\text{ order of elements = }2\mid\text{ $\#$ of elem in class = }1\mid\text{ representative = }\mathcal{T}^3\\
\text{Class }5\mid\text{ order of elements = }2\mid\text{ $\#$ of
elem in
class = }9\mid\text{ representative = }\mathcal{B}.\mathcal{A}.\mathcal{T}\\
\text{Class }6\mid\text{ order of elements = }2\mid\text{ $\#$ of elem in class = }9\mid\text{ representative = }\mathcal{B}.\mathcal{A}\\
\text{Class }7\mid\text{ order of elements = }2\mid\text{ $\#$ of elem in class = }9\mid\text{ representative = }\mathcal{B}.\mathcal{T}\\
\text{Class }8\mid\text{ order of elements = }2\mid\text{ $\#$ of elem in class = }9\mid\text{ representative = }\mathcal{B}\\
\text{Class }9\mid\text{ order of elements = }3\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}^2.\mathcal{T}^2\\
\text{Class }10\mid\text{ order of elements = }3\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}^2.\mathcal{T}^4\\
\text{Class }11\mid\text{ order of elements = }3\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}^2\\
\text{Class }12\mid\text{ order of elements = }3\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{T}^2\\
\text{Class }13\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}^3.\mathcal{T}^2\\
\text{Class }14\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}^3.\mathcal{T}\\
\text{Class }15\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}^2.\mathcal{T}^3\\
\text{Class }16\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}^2.\mathcal{T}\\
\text{Class }17\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}^2.\mathcal{T}^5\\
\text{Class }18\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}.\mathcal{T}^3\\
\text{Class }19\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}.\mathcal{T}^2\\
\text{Class }20\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}.\mathcal{T}^4\\
\text{Class }21\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}.\mathcal{T}\\
\text{Class }22\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}.\mathcal{T}^5\\
\text{Class }23\mid\text{ order of elements = }6\mid\text{ $\#$ of elem in class = }2\mid\text{ representative = }\mathcal{A}\\
\text{Class }24\mid\text{ order of elements = }6\mid\text{ $\#$ of
elem in class = }2\mid\text{ representative = }\mathcal{T}\)
\end{doublespace}
\subsubsection{Irreducible representations and the character table of
$\mathfrak{U}_{72}$} According with general theorems and with the
fact that $\mathfrak{U}_{72}$ is a solvable group we arrive at the
conclusion that it  has 24 irreps of which 8 are 1-dimensional and
16 are 2-dimensional. These representations are explicitly computed
once for all in a proper MATHEMATICA NoteBook which  created the
necessary files in the library to be uploaded while running the
other bckground NoteBook \textbf{HexagBckgroundN4} (see section
\ref{curucu}) dedicated to the construction of Beltrami flows and
then Navier Stokes solution associated with a spherical orbit of
momentum space. The character table of 24 irreps is given in table
\ref{curfucio}.
\begin{landscape}
\begin{table}
{{\fontsize{4.3}{4.8}
\begin{eqnarray*}\label{curuttero}
    \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
 D|C & \mathcal{C}_1 & \mathcal{C}_2 & \mathcal{C}_3 & \mathcal{C}_4 & \mathcal{C}_5 & \mathcal{C}_6 & \mathcal{C}_7 & \mathcal{C}_8 & \mathcal{C}_9
& \mathcal{C}_{10} & \mathcal{C}_{11} & \mathcal{C}_{12} &
\mathcal{C}_{13} & \mathcal{C}_{14} & \mathcal{C}_{15} &
\mathcal{C}_{16} & \mathcal{C}_{17}
& \mathcal{C}_{18} & \mathcal{C}_{19} & \mathcal{C}_{20} & \mathcal{C}_{21} & \mathcal{C}_{22} & \mathcal{C}_{23} & \mathcal{C}_{24} \\
\hline
 \text{Rep} & \mathcal{E} & \mathcal{A}^3.\mathcal{T}^3 & \mathcal{A}^3 & \mathcal{T}^3 & \mathcal{B}.\mathcal{A}.\mathcal{T} & \mathcal{B}.\mathcal{A}
& \mathcal{B}.\mathcal{T} & \mathcal{B} &
\mathcal{A}^2.\mathcal{T}^2 & \mathcal{A}^2.\mathcal{T}^4 &
\mathcal{A}^2 & \mathcal{T}^2 & \mathcal{A}^3.\mathcal{T}^2 &
\mathcal{A}^3.\mathcal{T} & \mathcal{A}^2.\mathcal{T}^3 &
\mathcal{A}^2.\mathcal{T} & \mathcal{A}^2.\mathcal{T}^5 &
\mathcal{A}.\mathcal{T}^3 &
\mathcal{A}.\mathcal{T}^2 & \mathcal{A}.\mathcal{T}^4 & \mathcal{A}.\mathcal{T} & \mathcal{A}.\mathcal{T}^5 & \mathcal{A} & \mathcal{T} \\
\hline
 D_1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
 D_2 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 \\
\hline
 D_3 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 \\
\hline
 D_4 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & 1 & -1 & -1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\
\hline
 D_5 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
 D_6 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 \\
\hline
 D_7 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 \\
\hline
 D_8 & 1 & 1 & -1 & -1 & -1 & 1 & 1 & -1 & 1 & 1 & 1 & 1 & -1 & 1 & -1 & -1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\
\hline
 D_9 & 2 & -2 & 2 & -2 & 0 & 0 & 0 & 0 & -1 & -1 & 2 & -1 & -1 & 1 & -2 & 1 & 1 & -2 & -1 & -1 & 1 & 1 & 2 & 1 \\
\hline
 D_{10} & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & -1 & -1 & 2 & -1 & -1 & -1 & 2 & -1 & -1 & 2 & -1 & -1 & -1 & -1 & 2 & -1 \\
\hline
 D_{11} & 2 & -2 & -2 & 2 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 2 & -2 & -2 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 \\
\hline
 D_{12} & 2 & 2 & -2 & -2 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & -1 & 1 & -1 & 1 & -2 & 1 & -1 & -2 & 1 & -1 & 2 & 1 & 1 \\
\hline
 D_{13} & 2 & -2 & -2 & 2 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & -1 & 1 & 1 & -1 & -1 & 2 & 1 & 1 & -2 & -2 & 1 & 1 & -1 \\
\hline
 D_{14} & 2 & 2 & -2 & -2 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 2 & -2 & 2 & 1 & 1 & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -2 \\
\hline
 D_{15} & 2 & -2 & -2 & 2 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & -1 & 1 & 1 & -1 & 2 & -1 & 1 & -2 & 1 & 1 & -2 & 1 & -1 \\
\hline
 D_{16} & 2 & 2 & -2 & -2 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & -1 & 1 & -1 & 1 & 1 & -2 & -1 & 1 & -2 & 2 & -1 & 1 & 1 \\
\hline
 D_{17} & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 2 & 2 & 2 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 2 \\
\hline
 D_{18} & 2 & -2 & 2 & -2 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & -2 & 1 & -1 & 2 & -2 & 1 & -1 & 1 \\
\hline
 D_{19} & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & -1 & -1 & -1 & -1 & 2 & -1 & -1 & 2 & -1 & -1 & 2 & -1 & -1 \\
\hline
 D_{20} & 2 & -2 & 2 & -2 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 2 & 2 & -2 & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -2 \\
\hline
 D_{21} & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 2 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 2 & -1 & -1 & 2 & 2 & -1 & -1 & -1 \\
\hline
 D_{22} & 2 & -2 & 2 & -2 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & -1 & -1 & 1 & 1 & -2 & 1 & 1 & 2 & -1 & 1 & -2 & -1 & 1 \\
\hline
 D_{23} & 2 & 2 & -2 & -2 & 0 & 0 & 0 & 0 & -1 & -1 & 2 & -1 & 1 & -1 & -2 & 1 & 1 & 2 & 1 & 1 & -1 & -1 & -2 & 1 \\
\hline
 D_{24} & 2 & -2 & -2 & 2 & 0 & 0 & 0 & 0 & -1 & -1 & 2 & -1 & 1 & 1 & 2 & -1 & -1 & -2 & 1 & 1 & 1 & 1 & -2 & -1 \\
\hline
\end{array}
\end{eqnarray*}}}
\caption{\scriptsize Character table of the $\mathfrak{U}_{72}$
group \label{curfucio}}
\end{table}
\end{landscape}
\section{Classification of orbits of the point group $\mathrm{Dih_6}$ in the momentum lattice}
In complete analogy with what it was done for the cubic lattice also
in the case of the hexagonal lattice we need to classify the orbits
of the point group $\mathrm{Dih_6}$ in the lattice
$\Lambda^\star_{Hex}$. Here we have six different types of orbits:
\subsection{Orbits of length 2}
These are the simplest orbits and are formed by vectors of the
following type:
\begin{eqnarray}\label{grongo2}
    \mathcal{O}_2 & = & \left\{p \, \lambda_3 \, , \, -p \,\lambda_3
    \right\} \quad ; \quad p \, \in \, \mathbb{Z} \nonumber\\
    &\Downarrow & \nonumber\\
    & = &\left\{\{0,0, \frac{p}{\sqrt{2}}\} \, , \,\{0,0,
    -\frac{p}{\sqrt{2}}\}
    \right\} \quad \text{in the orthonormal basis}
\end{eqnarray}
that are arranged along the $z$-axis. The action of the
$\mathcal{A}$ generator of the dihedral group vanishes on such
vectors and they are sensitive only to the $\mathcal{B}$ generators
that flips their orientation.
\subsection{Orbits of length 6}
The orbit of length 6 lies in the plane $z=0$ and are made by vectors
of the following type:
\begin{eqnarray}\label{grongo6}
    \mathcal{O}_6 & = & \underbrace{\left\{
\begin{array}{lcr}
 \{0, & -p, & 0 \}\\
\{ 0, & p, & 0\} \\
\{ -p, & 0, & 0 \}\\
 \{-p, & p, & 0\} \\
\{ p, & -p, & 0\} \\
\{ p, & 0, & 0\} \\
\end{array}
\right\}}_{\text{in the $\lambda_i$ basis}} \, = \, \underbrace{\left\{
\begin{array}{lcr}
 \{0, & -\sqrt{\frac{2}{3}} p, & 0 \}\\
 \{0, & \sqrt{\frac{2}{3}} p, & 0 \}\\
 \{-\frac{p}{\sqrt{2}}, & -\frac{p}{\sqrt{6}}, & 0\} \\
 \{-\frac{p}{\sqrt{2}}, & \frac{p}{\sqrt{6}}, & 0 \} \\
 \{\frac{p}{\sqrt{2}}, & -\frac{p}{\sqrt{6}}, & 0 \} \\
 \{\frac{p}{\sqrt{2}}, & \frac{p}{\sqrt{6}}, & 0\} \\
\end{array}
\right\}}_{\text{in the orthonormal basis}}\quad ; \quad p\, \in \,
\mathbb{Z}
\end{eqnarray}
See fig. \ref{duepiusei}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=50mm]{HexOrbit8Front.jpg}
\includegraphics[height=50mm]{HexOrbit8Abov.jpg}
\end{center}
\caption{\it  In this picture we mark with a big circle the points
in the hexagonal momentum lattice $\Lambda^\star_{Hex}$ that
constitute the lowest lying orbits of length 2 and 6. The orbit of
length 2 is given by the two marked antipodal points along the
vertical $z$-axis, while the orbit of length 6 is given by six
vertices of the regular hexagon lying in the horizontal plane $z=0$.
In the picture on the left we have a view of the lattice from the
front, in the picture on the right we have a view from above. The
blue points belong to the space lattice, the red points belong to
the dual momentum lattice.} \label{duepiusei}
\end{figure}
\subsection{Orbits of length 12 of type 1}
The orbits of length 12 and typo 1 lies in the $z=0$ plane and are of
the following form:
\begin{eqnarray}
\label{grongo12t1}
  \mathcal{O}_{12|1} &=& \underbrace{\left\{
\begin{array}{lcr}
 \{-p, & -q, & 0 \}\\
 \{-p, & p+q, & 0 \}\\
 \{p, & -p-q, & 0\} \\
\{ p, & q, & 0 \}\\
 \{-q, & -p, & 0\} \\
\{ -q, & p+q, & 0 \} \\
\{ q, & -p-q, & 0 \}\\
\{ q, & p, & 0 \}\\
\{ -p-q, & p, & 0 \}\\
\{ -p-q, & q, & 0 \}\\
\{ p+q, & -q, & 0 \} \\
\{ p+q, & -p, & 0 \} \\
\end{array}
\right\}}_{\text{in the $\lambda_i$ basis}} \, = \,\underbrace{\left\{
\begin{array}{lcr}
 \{-\frac{p}{\sqrt{2}}, & -\frac{p+2 q}{\sqrt{6}}, & 0 \}\\
\{ -\frac{p}{\sqrt{2}}, & \frac{p+2 q}{\sqrt{6}}, & 0 \}\\
\{ \frac{p}{\sqrt{2}}, & -\frac{p+2 q}{\sqrt{6}}, & 0 \}\\
\{ \frac{p}{\sqrt{2}} & \frac{p+2 q}{\sqrt{6}} & 0 \}\\
\{ -\frac{q}{\sqrt{2}}, & -\frac{2 p+q}{\sqrt{6}}, & 0 \}\\
\{ -\frac{q}{\sqrt{2}}, & \frac{2 p+q}{\sqrt{6}}, & 0 \}\\
\{ \frac{q}{\sqrt{2}}, & -\frac{2 p+q}{\sqrt{6}}, & 0 \}\\
\{ \frac{q}{\sqrt{2}}, & \frac{2 p+q}{\sqrt{6}}, & 0 \}\\
\{ -\frac{p+q}{\sqrt{2}}, & \frac{p-q}{\sqrt{6}}, & 0 \}\\
\{ -\frac{p+q}{\sqrt{2}}, & \frac{q-p}{\sqrt{6}}, & 0 \}\\
\{ \frac{p+q}{\sqrt{2}}, & \frac{p-q}{\sqrt{6}}, & 0 \}\\
\{ \frac{p+q}{\sqrt{2}}, & \frac{q-p}{\sqrt{6}}, & 0 \}\\
\end{array}
\right\}}_{\text{in the orthonormal basis}} \quad ; \quad p,q \, \in
\, \mathbb{Z}
\end{eqnarray}
\subsection{Orbits of length 12 of type 2}
The orbits of length 12 and type 2 are the most generic ones that
depend on three integers $p,q,r$ with no relation among them capable
of nullify some of the orthonormal components of the vectors
belonging to the orbit. Explicitly we find:
\begin{eqnarray}
\label{grongo12t2}
  \mathcal{O}_{12|2} &=& \underbrace{\left(
\begin{array}{lll}
\{-p , & -q , & r\} \\
\{-p , & p+q , & -r\} \\
\{ p , & -p-q , & -r\} \\
\{ p , & q , & r\} \\
\{-q , & -p , & -r\} \\
\{-q , & p+q , & r\} \\
\{q , & -p-q , & r\} \\
\{ q , & p , & -r\} \\
\{ -p-q , & p , & r\} \\
\{ -p-q , & q , & -r\} \\
\{ p+q , & -q , & -r\} \\
\{ p+q , & -p , & r\} \\
\end{array}
\right)}_{\text{in the $\lambda_i$ basis}} \, =
\,\underbrace{\left\{
\begin{array}{lll}
 \{-\frac{p}{\sqrt{2}} , & -\frac{p+2 q}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
 \{-\frac{p}{\sqrt{2}} , & \frac{p+2 q}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{\frac{p}{\sqrt{2}} , & -\frac{p+2 q}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{\frac{p}{\sqrt{2}} , & \frac{p+2 q}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
 \{-\frac{q}{\sqrt{2}} , & -\frac{2 p+q}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{-\frac{q}{\sqrt{2}} , & \frac{2 p+q}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
 \{\frac{q}{\sqrt{2}} , & -\frac{2 p+q}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
 \{\frac{q}{\sqrt{2}} , & \frac{2 p+q}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{-\frac{p+q}{\sqrt{2}} , & \frac{p-q}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
 \{-\frac{p+q}{\sqrt{2}} , & \frac{q-p}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{\frac{p+q}{\sqrt{2}} , & \frac{p-q}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{\frac{p+q}{\sqrt{2}} , & \frac{q-p}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
\end{array}
\right\}}_{\text{in the orthonormal basis}}
\end{eqnarray}
The parameters in the above orbit must satisfy the following
conditions:
\begin{equation}\label{condorbita2}
 p,q,r \,
\in \, \mathbb{Z} \quad \text{and} \quad
\left \{\begin{array}{ccc}q&\neq &- 2p \\ q &\neq & \pm p\\
\end{array}\right .
\end{equation}
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=50mm]{HexOrbit12Front.jpg}
\includegraphics[height=50mm]{HexOrbit12Abov.jpg}
\end{center}
\caption{\it  In this picture we mark with a big circle the points
in the hexagonal momentum lattice $\Lambda^\star_{Hex}$ that
constitute  an orbit of length 12 of type 2,3 or 4. The orbit of
length 12 is given by 12 vertices of a polyhedron which has two
opposite faces (upper and lower) corresponding to regular hexagons
on horizontal planes symmetrical under $z$-reflection and 6
rectangular vertical faces. In the picture on the left we have a
view of the lattice from the front, in the picture on the right we
have a view from above. The blue points belong to the space lattice,
the red points belong to the dual momentum lattice. The distinction
among type 2,3,4 depends only on the orientation of the hexagonal
faces in the lattice.} \label{dozzina}
\end{figure}
\subsection{Orbits of length 12 of type 3}
The orbits of length 12 and type 3 correspond to the degeneration of
the orbits of type 1 when $q=-p$. Explicitly we find:
\begin{eqnarray}
\label{grongo12t3}
  \mathcal{O}_{12|3} &=& \underbrace{\left\{
\begin{array}{ccc}
\{ 0 , & -p , & -r\} \\
\{ 0 , & -p , & r\} \\
\{ 0 , & p , & -r\} \\
\{ 0 , & p , & r\} \\
\{ -p , & 0 , & -r\} \\
\{ -p , & 0 , & r\} \\
\{ -p , & p , & -r\} \\
\{ -p , & p , & r\} \\
\{ p , & -p , & -r\} \\
\{ p , & -p , & r\} \\
\{ p , & 0 , & -r\} \\
\{ p , & 0 , & r\} \\
\end{array}
\right\}}_{\text{in the $\lambda_i$ basis}} \, =
\,\underbrace{\left\{
\begin{array}{lll}
 \{0 , & -\sqrt{\frac{2}{3}} p , & -\frac{r}{\sqrt{2}}\} \\
 \{0 , & -\sqrt{\frac{2}{3}} p , & \frac{r}{\sqrt{2}}\} \\
 \{0 , & \sqrt{\frac{2}{3}} p , & -\frac{r}{\sqrt{2}}\} \\
 \{0 , & \sqrt{\frac{2}{3}} p , & \frac{r}{\sqrt{2}}\} \\
 \{-\frac{p}{\sqrt{2}} , & -\frac{p}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{-\frac{p}{\sqrt{2}} , & -\frac{p}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
 \{-\frac{p}{\sqrt{2}} , & \frac{p}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{-\frac{p}{\sqrt{2}} , & \frac{p}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
 \{\frac{p}{\sqrt{2}} , & -\frac{p}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{\frac{p}{\sqrt{2}} , & -\frac{p}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
 \{\frac{p}{\sqrt{2}} , & \frac{p}{\sqrt{6}} , & -\frac{r}{\sqrt{2}}\} \\
 \{\frac{p}{\sqrt{2}} , & \frac{p}{\sqrt{6}} , & \frac{r}{\sqrt{2}}\} \\
\end{array}
\right\}}_{\text{in the orthonormal basis}} \quad ; \quad  p,r \,
\in \, \mathbb{Z}
\end{eqnarray}
\subsection{Orbits of length 12 of type 4}
The orbits of length 12 and type 4 correspond to the degeneration of
the orbits of type 1 when $q=-2p$. Explicitly we find:
\begin{eqnarray}
\label{grongo12t4}
  \mathcal{O}_{12|4} , &=, & \underbrace{\left\{
\begin{array}{lll}
\{ -p , & -p , & -r \}\\
\{ -p , & -p , & r \}\\
\{ -p , & 2 p , & -r \}\\
\{ -p , & 2 p , & r \}\\
\{ p , & -2 p , & -r\} \\
\{ p , & -2 p , & r \}\\
\{ p , & p , & -r \} \\
\{ p , & p , & r\} \\
\{ -2 p , & p , & -r\} \\
\{ -2 p , & p , & r\} \\
\{ 2 p , & -p , & -r\} \\
\{ 2 p , & -p , & r\} \\
\end{array}
\right\}}_{\text{in the $\lambda_i$ basis}} \, =
\,\underbrace{\left\{
\begin{array}{lll}
\{ -\frac{p}{\sqrt{2}} , & -\sqrt{\frac{3}{2}} p , & -\frac{r}{\sqrt{2}}\} \\
\{ -\frac{p}{\sqrt{2}} , & -\sqrt{\frac{3}{2}} p , & \frac{r}{\sqrt{2}}\} \\
\{ -\frac{p}{\sqrt{2}} , & \sqrt{\frac{3}{2}} p , & -\frac{r}{\sqrt{2}} \}\\
\{ -\frac{p}{\sqrt{2}} , & \sqrt{\frac{3}{2}} p , & \frac{r}{\sqrt{2}} \}\\
\{ \frac{p}{\sqrt{2}} , & -\sqrt{\frac{3}{2}} p , & -\frac{r}{\sqrt{2}}\} \\
\{ \frac{p}{\sqrt{2}} , & -\sqrt{\frac{3}{2}} p , & \frac{r}{\sqrt{2}}\} \\
\{ \frac{p}{\sqrt{2}} , & \sqrt{\frac{3}{2}} p , & -\frac{r}{\sqrt{2}}\} \\
\{ \frac{p}{\sqrt{2}} , & \sqrt{\frac{3}{2}} p , & \frac{r}{\sqrt{2}} \}\\
\{ -\sqrt{2} p , & 0 , & -\frac{r}{\sqrt{2}}\} \\
\{ -\sqrt{2} p , & 0 , & \frac{r}{\sqrt{2}}\} \\
\{ \sqrt{2} p , & 0 , & -\frac{r}{\sqrt{2}}\} \\
\{ \sqrt{2} p , & 0 , & \frac{r}{\sqrt{2}}\} \\
\end{array}
\right\}}_{\text{in the orthonormal basis}} \quad ; \quad p,r \, \in
\, \mathbb{Z}
\end{eqnarray}
See a picture of orbits of length 12 of type 2,3,4 in
fig.\ref{dozzina}
As we see the shortest orbit of length $2$ is actually vertical,
namely the associated Beltrami Flows correspond to decoupled systems
where only the coordinate $z(t)$ obeys a non linear differential
equation. The other two coordinates form a free system. Similarly
the orbits of length $6$ and the first orbit of length $12$ are all
planar. In the corresponding Beltrami Flows there is no dependence
on the coordinate $z$ which forms a free system. Presumably all the
Beltrami Flows of this type are integrable. Only the maximal orbits
of length $12$  of type two, three and four are truly
three-dimensional and give rise to systems that might develop chaos.
\chapter{Group Theory and $\mathfrak{b}$-Beltrami fields}
\label{dippo}
\section{The Euler equations in a $\mathfrak{b}$-three-manifold and
the ABC model as a test ground}\label{bABCfildo} In view of the
geometrical setup discussed in chapter \ref{bmanstory}, in the
present one we reconsider Euler equations and Beltrami fields in
$\mathfrak{b}$-manifolds, following the approach of
\cite{cardone2019} and focusing in particular on the example of the
$ABC$-flows that they used there. Our aim is to bring up to evidence
the relation existing between the necessary condition found in
\cite{cardone2019} for the consistency of Beltrami equation in a
particular $b$-manifold with a particular boundary surface $\Sigma$
and the group theoretical structure of the ABC-model that was
exhaustively presented in \cite{Fre:2015mla}. We will argue that
such a relation is most likely general and that the possible types
of boundary surfaces $\Sigma$ which can be introduced in Beltrami
fields is to be classified in group theoretical terms also in the
case of the much more complicated Beltrami flows originated by
higher orbits of the point-group in the momentum lattice.
\subsection{The appropriate geometrical rewriting of Euler equations
on general three-manifolds} In order to implement our programme we
come once again back to Euler equation as written in eq.
(\ref{bernullone}) which, in view of the definition of the Bernoulli
function given in eq.(\ref{finocchionabiscotta}) and for steady
flows can be stated as follows:
\begin{equation}\label{grammomole}
    i_{\mathbf{U}}\cdot \mathrm{d}\Omega^{\mathbf{U}} \, =\, - \mathrm{d}H_B
\end{equation}
We remind the reader that, geometrically, the one-form
$\Omega^{\mathbf{U}}$ is the \textit{contact form}, the velocity
field $\mathbf{U}$ is its \textit{Reeb-field} and $H_B$ is indeed
the Bernoulli-function. In addition to eq. (\ref{grammomole}) the
dynamical system requires, in order to be complete, the
divergenceless condition:
\begin{equation}\label{nonscappo}
   \nabla\cdot \mathbf{U} \, \, \equiv \, \frac{1}{\sqrt{\text{det} g}}
  \, \partial_\ell \left(\sqrt{\text{det} g}\, U^\ell\right) \, = \, 0
\end{equation}
where $g_{ij}$ is the metric tensor of the three-manifold. Also
equation (\ref{nonscappo}) admits an index-free totally geometrical
rewriting in terms of the volume three-form defined belove:
\begin{equation}\label{lopezdevega}
    \text{Vol}_g \, \equiv \, \frac{1}{3!} \, \sqrt{\text{det} g} \,
    \epsilon_{ijk} \, dx^i \wedge dx^j \wedge dx^k
\end{equation}
An easy straightforward calculations shows that:
\begin{equation}\label{speusippo}
    \mathrm{d} \left( i_{\mathbf{U}}\cdot\text{Vol}_g \right) \, = \, \nabla\cdot
    \mathbf{U} \, \times \, \text{Vol}_g
\end{equation}
Hence Euler equations reduce to:
\begin{eqnarray}
\label{eulerazna}
  i_{\mathbf{U}}\cdot \mathrm{d}\Omega^{\mathbf{U}} & =& - \mathrm{d}H_B \nonumber\\
  \mathrm{d} \left( i_{\mathbf{U}}\cdot\text{Vol}_g \right) &=& 0
\end{eqnarray}
\subsection{The $b$-deformation of the ABC-model}
\label{bABCsekzia} Next we consider the ABC model vector field as
defined in the next section in eq.(\ref{ABCnostro}) and we try
convert the $\mathrm{T^3}$ torus, obtained by quotiening
$\mathbb{R}^3$ with respect to the cubic lattice, into a
$\mathfrak{b}$-manifold by choosing, in the covering space
$\mathbb{R}^3$, the surface $\Sigma_{x=0}\subset \mathbb{R}^3$
identified by the equation $x=0$. The Beltrami vector becomes
parallel to the surface $\Sigma_{x=0}$ by means of the substitution:
\begin{equation}\label{bigiazione}
    \partial_x \, \longrightarrow \, x\, \partial_x
\end{equation}
Hence we have:
\begin{eqnarray}\label{pariola}
    \null^{\mathfrak{b}}\mathbf{V}_{ABC}&=&\left(2 A \cos [2 \pi  y]+2 B \cos [2 \pi  z]\right)\,x
    \partial_x\nonumber\\
   &&(2 C \cos [2 \pi  x]-2 B \sin [2 \pi  z])\,\partial_y \,  +\,\left(2 A \sin [2 \pi  y]-2 C
   \sin [2 \pi  x]\right)\,\partial_z
\end{eqnarray}
According to the principles of $\mathfrak{b}$-manifolds summarized
in in section \ref{bmanstory} the metric and the differential forms
are accordingly modified. We have:
\begin{eqnarray}
\label{perlafiorita}
\null^{\mathfrak{b}}ds^2 &=& \null^{\mathfrak{b}}g_{ij}dx^i \times dx^j \, = \, \left(\frac{dx}{x}\right)^2 + dy^2 +dz^2\nonumber \\
\null^{\mathfrak{b}}\text{Vol}_{g} &=& \frac{dx}{x}\wedge dy \wedge
dz
\end{eqnarray}
so we easily compute:
\begin{eqnarray}\label{girdiniere}
    i_{\null^{\mathfrak{b}}\mathbf{V}_{ABC}}\cdot\null^{\mathfrak{b}}\text{Vol}_{g}&
    = & 2 dy\wedge dz \left(A \cos [2 \pi  y]+B \cos [(2 \pi  z]\right)\nonumber\\
    &&-\frac{2 dx\wedge
   dy \left(C \sin [2 \pi  x]-A \sin [2 \pi  y]\right)}{x}
   -\frac{2 dx\wedge dz \left(C
   \cos [2 \pi  x]-B \sin [2 \pi  z]\right)}{x}\nonumber\\
\end{eqnarray}
and we immediately verify the second of eq.s (\ref{eulerazna})
\begin{equation}\label{bnonscappo}
    \mathrm{d}\left(i_{\null^{\mathfrak{b}}\mathbf{V}_{ABC}}\cdot\null^{\mathfrak{b}}\text{Vol}_{g}\right)\,
    = \, 0
\end{equation}
As we know from the discussion in the introduction the first of
eq.s(\ref{eulerazna}) this certainly satisfied if the stronger
Beltrami equation (\ref{Belatramus}) is enforced and before the
$\mathfrak{b}$-deformation the contact form
$\Omega^{\left[\mathbf{V}_{ABC}\right]}$ certainly satisfies it by
construction. It is to be seen whether the new
$\mathfrak{b}$-contact form
$\null^{\mathfrak{b}}\Omega^{\left[\mathbf{V}_{ABC}\right]}$ still
satisfies it. We easily compute:
\begin{eqnarray}\label{bicontattoOmABC}
   \null^{\mathfrak{b}}\Omega^{\left[\mathbf{V}_{ABC}\right]}\,
   =\,\null^{\mathfrak{b}}g_{ij}\null^{\mathfrak{b}}\mathbf{V}_{ABC}^i
   \, dx^j & = & \frac{dx \left(2 A \cos [2 \pi  y]+2 B \cos [2 \pi  z]\right)}{x}\nonumber\\
   &&+dy \left(2 C \cos [2 \pi  x]-2 B \sin [2 \pi
   z]\right)+dz \left(2 A \sin [2
   \pi  y]-2 C \sin [2 \pi  x]\right)\nonumber\\
\end{eqnarray}
Taking the Hodge dual of Beltrami equation (\ref{Belatramus}) we can
equivalently rewrite it as follows:
\begin{equation}\label{machiavellico}
    \mathrm{d}\null^{\mathfrak{b}}\Omega^{\left[\mathbf{V}_{ABC}\right]}\,
    = \, \lambda \, \left(\star_{\,\null^{\mathfrak{b}}g}\,
    \null^{\mathfrak{b}}\Omega^{\left[\mathbf{V}_{ABC}\right]} \right) \,
    \equiv
    \, \lambda \, \ft 12 \left(
    i_{\null^{\mathfrak{b}}\mathbf{V}_{ABC}}\cdot\null^{\mathfrak{b}}\text{Vol}_{g}\right)
\end{equation}
The second member was already calculated, the first is immediately
calculated and we find that setting $\lambda\, = \, 2\pi$ which is
its original value prior to the deformation, we have:
\begin{equation}\label{kunago}
    \mathrm{d}\null^{\mathfrak{b}}\Omega^{\left[\mathbf{V}_{ABC}\right]}\,-\,
    \pi \,\left(
    i_{\null^{\mathfrak{b}}\mathbf{V}_{ABC}}\cdot\null^{\mathfrak{b}}\text{Vol}_{g}\right)
    \, = \,-\frac{4 \pi  C (x-1) \left(\sin [2 \pi  x] dx\wedge dy+\cos [2 \pi
    x]
   dx\wedge dz\right)}{x}
\end{equation}
As it was done in \cite{cardone2019} we have no other way out then
choosing $C=0$. Hence we conclude that the complete ABC-model cannot
be $\mathfrak{b}$-deformed but the AB0-model can. In
\cite{cardone2019} the boundary surface was posed at $z=0$ and the
authors reached the same conclusion in the form of the constraint
$A=0$. As we argue in the next section these two choices are
perfectly equivalent since we can interchange $A,B,C$ with
transformations of the subgroup $\mathrm{GF_{192}}\subset
\mathrm{G_{1536}}$ of which the ABC model constitutes an irreducible
three dimensional representation. The important thing is that
setting one of the three parameters ABC, equal to zero one obtains a
two-parameter model which constitutes an irreducible representation
of a subsgroup $\mathrm{G_{128}^{(AB0)}}\subset
\mathrm{G_{1536}}$\footnote{The subgroups $\mathrm{G_{128}^{(0BC)}}$
and $\mathrm{G_{128}^{(0BC)}}$ are obviously conjugate in $G_{1536}$
to $\mathrm{G_{128}^{(AB0)}}$ and therefore isomorphic  to this
latter and among themselves}. Inside $\mathrm{G_{128}^{(AB0)}}$ the
stabilizer of the two vector $(A,B)$ is a group of order 16
$G^{(AB0)}_{16}$ which contains a purely translational subgroup
$G^{(AB0)}_4 \sim \mathbb{Z}_4$ made by the quantized translation of
$1/4$ in the $y$-direction. This makes the dynamical system actually
two-dimensional. It is a remarkable fact that the
$\mathfrak{b}$-deformation of the chosen type is possible only in
presence of this particular hidden symmetry. We come back to this
question at the end of the chapter. First we recall from
\cite{Fre:2015mla} the group-theory behind the ABC models.
\section{Group theoretical interpretation of the
ABC flows} From the  analysis \cite{Fre:2015mla}  the following
pattern emerged. The Universal Classifying Group $\mathrm{G_{1536}}$
contains at least two\footnote{It is known that there are 4
different Space-Groups $\Gamma^{I}_{24}$ ($I=1,\dots,4$) of order
$24$, isomorphic to the point group $\mathrm{O_{24}}$ but not
conjugate one to the other under the action of the continuous
translation group. One of them is the point group  itself
$\Gamma^{1}_{24}\, = \, \mathrm{O_{24}}$ which is a subgroup of the
first of the two groups of order 192 identified in
\cite{Fre:2015mla}: $\mathrm{O_{24}} \subset \mathrm{G_{192}}$.
Another of the four mentioned groups is $\Gamma^{2}_{24}\, = \,
\mathrm{GS_{24}}$ which is a subgroup of the second group of order
192 identified by the authors of \cite{Fre:2015mla}:
$\mathrm{GS_{24}} \subset \mathrm{GF_{192}}$. It remains to see
whether $\Gamma^{3}_{24} $ and $ \Gamma^{4}_{24}$ are contained in
the two already identified subgroups $\mathrm{G_{192}}$ and
$\mathrm{GF_{192}}$ or if there exists other two such non conjugate
subgroups of order 192 that respectively contain $\Gamma^{3}_{24} $
and $\Gamma^{3}_{24} $. The answer was not worked out in
\cite{Fre:2015mla}.  Extensive but lengthy calculations could
resolve the issue.} isomorphic but not conjugate subgroups of order
192, namely $\mathrm{G_{192}}$ and $\mathrm{GF_{192}}$ in the
adopted nomenclature. The classical ABC-flows are obtained from the
lowest lying momentum orbit of length 6 which produces an
irreducible $6$-dimensional representation of the Universal
Classifying Group: $D_{23}\left[\mathrm{\mathrm{G_{1536}}},6\right]
$. The vector field is the following one:
\begin{equation}\label{orbit6vector}
{\mathbf{V}^{(6)}}({\mathbf{r}}|\mathbf{F})\,  = \,\left(
\begin{array}{l}
 2 \cos \left(2 \pi  z\right) F_1+2 \cos \left(2 \pi  y\right)
   F_2+2 \sin \left(2 \pi  z\right) F_3-2 \sin \left(2 \pi  \Theta
   _2\right) F_5 \\
 -2 \sin \left(2 \pi  z\right) F_1+2 \cos \left(2 \pi  z\right)
   F_3+2 \cos \left(2 \pi  x\right) F_4+2 \sin \left(2 \pi  \Theta
   _1\right) F_6 \\
 2 \sin \left(2 \pi  y\right) F_2-2 \sin \left(2 \pi  x\right)
   F_4+2 \cos \left(2 \pi  y\right) F_5+2 \cos \left(2 \pi  \Theta
   _1\right) F_6
\end{array}
\right)
\end{equation}
where $F_i$ ($i\,=\,1,\dots\, ,6$) are real numbers. The three
parameter ABC-flow is just the irreducible $3$-dimensional
representation $D_{12}\left[\mathrm{GF_{192}},3\right]$ in the split
\begin{equation}\label{splittus}
    D_{23}\left[\mathrm{\mathrm{G_{1536}}},6\right] \, = \,
D_{12}\left[\mathrm{GF_{192}},3\right]\oplus
D_{15}\left[\mathrm{GF_{192}},3\right]
\end{equation}
 With respect to the
isomorphic but not conjugate subgroup $\mathrm{G_{192}}$ the
representation $D_{23}\left[\mathrm{\mathrm{G_{1536}}},6\right] $
remains instead irreducible:
\begin{equation}\label{nosplittus}
    D_{23}\left[\mathrm{\mathrm{G_{1536}}},6\right] \, = \,
D_{20}\left[\mathrm{G_{192}},6\right]
\end{equation}
so that there is no proper way of reducing the six parameters to
three.
\par
Indeed, as shown in \cite{Fre:2015mla}, we have the following chain
of inclusions:
\begin{equation}\label{baldacchinoA}
    \mathrm{\mathrm{G_{1536}}} \, \supset \, \mathrm{GF_{192}} \, \supset \, \mathrm{GS_{24}}
\end{equation}
that is parallel to the other one:
\begin{equation}\label{baldacchinoB}
    \mathrm{\mathrm{G_{1536}}} \, \supset \, \mathrm{G_{192}} \, \supset \, \mathrm{O_{24}}
\end{equation}
$\mathrm{G_{192}}$ being another subgroup, isomorphic to
$\mathrm{GF_{192}}$, but not conjugate to it in
$\mathrm{\mathrm{G_{1536}}}$. (see appendices A.6 and A.7 of
\cite{Fre:2015mla} for the detailed description of these two
subgroups of the Universal Classifying Group $\mathrm{G_{1536}}$ of
the cubic lattice). Since $\mathrm{G_{192}}$ and $\mathrm{GF_{192}}$
are isomorphic they have the same irreps and the same character
table. Yet, since they are not conjugate, the branching rules of the
same $\mathrm{\mathrm{G_{1536}}}$ irrep with respect to the former
or the latter subgroup can be different. In the case of the
representation $D_{23}\left[\mathrm{\mathrm{G_{1536}}},6\right]$,
which is that produced by the fundamental orbit of order six, we
have (see appendix D of \cite{Fre:2015mla}):
\begin{equation}\label{passerotto}
D_{23}\left[\mathrm{\mathrm{G_{1536}}},6\right] \, = \,
\left\{\begin{array}{rcl}
D_{20}\left[\mathrm{G_{192}},6 \right] &=&D_4\left[\mathrm{O_{24}},3\right]  \oplus D_5\left[\mathrm{O_{24}},3\right]\\
D_{12}\left[\mathrm{GF_{192}},3 \right]  \oplus
D_{15}\left[\mathrm{GF_{192}},3
\right]&=&D_1\left[\mathrm{GS_{24}},1\right]
\oplus  D_3\left[\mathrm{GS_{24}},2\right] \oplus  D_4\left[\mathrm{GS_{24}},3\right]\\
\end{array}
\right.
\end{equation}
where in the second line we have used the branching rules:
\begin{eqnarray}
D_{12}\left[\mathrm{GF_{192}},3 \right]  &=& D_1\left[\mathrm{GS_{24}},1\right] \oplus  D_3\left[\mathrm{GS_{24}},2\right]  \\
D_{15}\left[\mathrm{GF_{192}},3 \right] &=&
D_4\left[\mathrm{GS_{24}},3\right] \label{passamontagna}
\end{eqnarray}
that, in view of the isomorphism, are identical with:
\begin{eqnarray}
 D_{12}\left[\mathrm{G_{192}},3 \right]  &=& D_1\left[\mathrm{O_{24}},1\right] \oplus  D_3\left[\mathrm{O_{24}},2\right]  \\
  D_{15}\left[\mathrm{G_{192}},3 \right] &=&  D_4\left[\mathrm{O_{24}},3\right] \label{passamontagnabis}
\end{eqnarray}
Eq.(\ref{passerotto}) has far reaching consequences. While there are
no Beltrami vector fields obtained from this orbit that are
invariant with respect to the octahedral point group
$\mathrm{O_{24}}$, there exists such an invariant Beltrami flow with
respect to the isomorphic $\mathrm{GS_{24}}$: it corresponds to the
$D_1\left[\mathrm{GS_{24}},1\right]$ irrep in the second line of
(\ref{passerotto}). Furthermore while the six parameter space
$\mathbf{F}$ is irreducible with respect to the action of the group
$\mathrm{G_{192}}$ (the irrep $D_{20}\left[\mathrm{G_{192}},6
\right]$) it splits into two three-dimensional subspaces with
respect to $\mathrm{GF_{192}}$. This is the origin of the ABC-flows.
Indeed the ABC Beltrami flows can be identified with the irreducible
representation $ D_{12}\left[\mathrm{GF_{192}},3 \right] $. Let us
see how. Explicitly we have the following projection operators on
the two irreducible representations,
 $D_{12}$ and $D_{15}$:
\begin{eqnarray}
  \Pi^{(12)}\left[\mathrm{GF_{192}},3\right]\,\mathbf{F}  &=& \left\{F_1,F_2,0,F_4,0,0\right\} \\
  \Pi^{(15)}\left[\mathrm{GF_{192}},3\right]\,\mathbf{F}&=& \left\{0,0,F_3,0,F_5,F_6\right\}
\end{eqnarray}
If we set $F_3\,=\,F_5\, = \, F_6=0$ we kill the irreducible
representation $ D_{15}\left[\mathrm{GF_{192}},3 \right] $ and the
residual Beltrami vector field, upon the following identifications:
\begin{equation}\label{agnorizo}
   A\, = \, F_1 \quad ; \quad B\, = \, F_4 \quad ; \quad C \, = \, F_2
\end{equation}
coincides with the time honored ABC flow of eq.(\ref{bagcigaluppi}).
Indeed inserting  the special parameter vector $\mathbf{F}\, = \,
\{A,C,0,B,0,0\}$ in eq.(\ref{orbit6vector}) we obtain:
\begin{eqnarray}\label{sicumerulo}
\mathbf{V}^{(6)}\left(\left\{x,y,z\right\}\,
\mid\{A,C,0,B,0,0\}\right)& \equiv \,
&\mathbf{V}_{(ABC)}(x,y,z)\nonumber\\
\mathbf{V}_{(ABC)}\left(x+\ft 34,y,z-\ft
14\right)&=&\mathcal{V}^{(ABC)}(x,y,z)
\end{eqnarray}
the vector field $\mathcal{V}^{(ABC)}(x,y,z)$ being that defined by
eq.(\ref{bagcigaluppi}).
\par
For future quick reference it is convenient to write explicitly the
ABC Beltrami field $\mathbf{V}^{(ABC)}(x,y,z)$ in the normalization
we utilize in the sequel:
\begin{equation}\label{ABCnostro}
    \mathbf{V}^{(ABC)}(x,y,z)\, = \, \left(
\begin{array}{c}
 2 A \cos (2 \pi  y)+2 B \cos (2 \pi  z) \\
 2 C \cos (2 \pi  x)-2 B \sin (2 \pi  z) \\
 2 A \sin (2 \pi  y)-2 C \sin (2 \pi  x) \\
\end{array}
\right)
\end{equation}
\par
The next step is provided by considering the explicit form of the
decomposition of the $ D_{12}\left[\mathrm{GF_{192}},3 \right] $
irrep, \textit{i.e.} the ABC flow, into irreducible representations
of the subgroup $\mathrm{GS_{24}}$. The two invariant subspaces are
immediately characterized in terms of the parameters $A,B,C$, as it
follows:
\begin{eqnarray}
 D_1\left[\mathrm{GS_{24}},1\right]  &\Leftrightarrow& A\, = \, B \, = \, C \, \ne \, 0 \\
 D_3\left[\mathrm{GS_{24}},2\right] &\Leftrightarrow& A\, + \, B \, + \, C \, = \, 0 \label{ergastolo}
\end{eqnarray}
The most symmetric case $A:A:A=1$ simply corresponds to the identity
representation of the subgroup  $\mathrm{GS_{24}}\subset
\mathrm{GF_{192}}$ which occurs in the splitting  of  the
$3$-dimensional representation:
\begin{equation}\label{GS24splitto}
   D_{12}\left[\mathrm{GF_{192}},3\right]\, = \,
D_{1}\left[\mathrm{GS_{24}},1\right]\oplus
D_{3}\left[\mathrm{GS_{24}},2\right]
\end{equation}
\subsection{The $\mathrm{(A,A,A)}$-flow invariant under
$\mathrm{GS_{24}}$} This information suffices to understand the role
of the $A:A:A\, =\,1$ Beltrami vector field often considered in the
literature. It is the unique one invariant under the order 24 group
$\mathrm{GS_{24}}$ isomorphic to the octahedral point group.
Explicitly, in our notations, it takes the following
form\footnote{Observe that here and in the sequel we stick to our
conventions for $x,y,z$, which differ from those of
eq.(\ref{bagcigaluppi}) by the already mentioned shift $\{ \ft 34 ,
0,-\ft 14\}$)}:
\begin{equation}\label{AAAfildo}
  \mathbf{V}_{(A,A,A)}(\mathbf{r})\, = \,   \mathbf{V}_{(A,A,A)}(x,y,z)\, \equiv \, 2 A \, \left(
\begin{array}{l}
  (\cos (2 \pi  y)+\cos (2 \pi  z)) \\
  (\cos (2 \pi  x)-\sin (2 \pi  z)) \\
 (\sin (2 \pi  y)-\sin (2 \pi  x))
\end{array}
\right)
\end{equation}
This vector field $\mathbf{V}_{(A,A,A)}(x,y,z)$ is everywhere non
singular in the fundamental unit cube (the torus $\mathrm{T^3}$)
apart from eight isolated   \textit{stagnation points} where it
vanishes. They are listed below.
\begin{equation}\label{gommaliqua}
\begin{array}{rclcrcl}
s_1 &= & \left\{\frac{1}{8},\frac{1}{8},\frac{3}{8}\right\} &; &
s_2 &= &\left\{\frac{1}{8},\frac{3}{8},\frac{1}{8}\right\}\\
s_3 &= & \left\{\frac{3}{8},\frac{1}{8},\frac{5}{8}\right\} &;&
s_4 &= &\left\{\frac{3}{8},\frac{3}{8},\frac{7}{8}\right\}\\
s_5 &= & \left\{\frac{5}{8},\frac{5}{8},\frac{7}{8}\right\} &;&
s_6&=&\left\{\frac{5}{8},\frac{7}{8},\frac{5}{8}\right\}\\
s_7&= &\left\{\frac{7}{8},\frac{5}{8},\frac{1}{8}\right\} &;& s_8 &
=& \left\{\frac{7}{8},\frac{7}{8},\frac{3}{8}\right\}
\end{array}
\end{equation}
A numerical plot of this vector field is displayed in fig.
\ref{AAAcamporella}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=90mm]{AAAfildus.png}
\end{center}
\caption{\it  A plot of the $A:A:A=1$ Beltrami vector field
invariant under the group $\mathrm{GS_{24}}$ with a view of its
eight stagnation points of eq.(\ref{gommaliqua})}
\label{AAAcamporella} \hskip 1cm \unitlength=1.1mm
\end{figure}
\par
In order to provide the reader with a visual impression of the
dynamics of this flow, in fig.\ref{AAAfiliderba} we display a set of
$5\times 5 \times 5 = 125$ streamlines, namely of numerical
integrations of the differential system:
\begin{equation}\label{babbione}
    \frac{dr}{dt} \, = \, \mathbf{V}_{(A,A,A)}(\mathbf{r})
\end{equation}
with initial conditions:
\begin{equation}\label{iniconda}
    \mathbf{r}(0) \, = \, \mathbf{r}_0 \, = \, \left\{\frac{n_1}{6},\, \frac{n_2}{6}, \,\frac{n_3}{6}\right\} \quad ;
    \quad n_{1,2,3} \, =\, 0,1,2,3,4,5
\end{equation}
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=90mm]{AAAfilini.png}
\end{center}
\caption{\it  A plot of $216$ streamlines of the $A:A:A=1$ Beltrami
vector field with equally spaced initial conditions. The numerical
solutions are smooth in $\mathbb{R}^3$. When a branch reaches a
boundary of the unit cube it is continued with its image in the cube
modulo the appropriate lattice translation. The circles in this
figure are the eight stagnation points } \label{AAAfiliderba}
\end{figure}
\section{Chains of subgroups and the flows $(A,B,0)$, $(A,A,0)$ and
$(A,0,0)$} In the literature a lot of attention has been given to
the special subcases of the $\mathrm{ABC}$-flow where one or two of
the parameters vanish or two are equal among themselves and one
vanishes. Also these cases can be thoroughly characterized in group
theoretical terms and their special features can be traced back to
the hidden subgroup structure associated with them.
\subsection{The
$(A,B,0)$ case and its associated chain of subgroups}
\label{AB0floppo} First we consider the case where we put to zero
one of the three parameters leaving the other two undetermined.
\par A preliminary important observation is the following. Each of
the three parameters is associated in eq.(\ref{ABCnostro}) with the
trigonometric functions of one of the three variables $x,y,z$. Hence
permuting the variables $x,y,z$ is equivalent to permute the $A,B,C$
coefficients. There are also some changes of sign but all these
operations are contained in the point group $\mathrm{O_{24}}$ as one
can immediately realize looking at eq.(\ref{nomiOelemen}). Hence in
the Universal Classifying Group $\mathrm{G_{1536}}$ that contains
the point group there are certainly elements that can map the
parameter vector $\{A,B,C\}$ in any other permutation of the same
letters. That means that considering the case $C=0$ is no loss of
generality. The invariance groups that we determine for this case
will just be conjugate to the invariance groups appearing in the
case $A=0$ or in the case $B=0$. So let us make the choice $C=0$
which was already done in \cite{Fre:2015mla}.
\paragraph{When we put $C=0$} we define a two dimensional subspace of the representation
$D_{12}\left[\mathrm{GF_{192}},3\right]$ which is invariant under
some proper subgroup $\mathrm{H}^{\mathrm{(A,B,0)}} \subset
\mathrm{GF_{192}}$. This group $\mathrm{H}^{\mathrm{(A,B,0)}}$ can
be calculated and found to be of order 64, yet we do not dwell on it
because the subgroup of the classifying group
$\mathrm{\mathrm{G_{1536}}}$ which leaves the subspace
$(A,0,0,B,0,0)$ invariant is larger than
$\mathrm{H}^{\mathrm{(A,B,0)}}$ and it is not contained in
$\mathrm{GF_{192}}$. It has order 128 and we name it
$\mathrm{G_{128}^{(A,B,0)}}$. This short discussion is important
because it implies the following: the flows $\mathrm{(A,B,0})$
should not be considered just as a particular case of the
$ABC$-flows rather as a different set of flows, whose properties are
encoded in the group $\mathrm{G_{128}^{(A,B,0)}}$.
\par
The group $\mathrm{G_{128}^{(A,B,0)}}$ is solvable and a chain of
normal subgroups can be found, all of index 2 which ends with the
abelian $\mathrm{G_{4}^{(A,B,0)}}$ isomorphic to $\mathbb{Z}_4$.
This latter is nothing else than the group of quantized translation
in the $y$-direction and its inclusion in the group leaving the
space $(A,0,0,B,0,0)$ invariant actually means that the differential
system must be  $y$-independent and hence two dimensional. The chain
of normal subgroups is displayed here below:
\begin{center}
\begin{picture}(200,100)
\put(-90,45){$\mathbb{Z}_4$} \put(-75,45){$\sim$} \put
(-60,45){$\mathrm{G_4^{(A,B,0)}}$} \put (-20,45){$\vartriangleleft
$} \put (-5,45){$\mathrm{G_8^{(A,B,0)}}$} \put
(35,45){$\vartriangleleft $} \put
(50,45){$\mathrm{G_{16}^{(A,B,0)}}$} \put (90,45){$\vartriangleleft
$} \put (103,47){\line (1,1){20}} \put (103,47){\line (1,-1){20}}
\put (127,65.5){$\vartriangleleft $} \put (127,24){$\vartriangleleft
$} \put (142,24){$\mathrm{G_{32}^{(A,A,0)}}$} \put
(142,65.5){$\mathrm{G_{32}^{(A,B,0)}}$} \put
(182,65.5){$\vartriangleleft $} \put
(197,65.5){$\mathrm{G_{64}^{(A,B,0)}}$} \put
(237,65.5){$\vartriangleleft $} \put
(252,65.5){$\mathrm{G_{128}^{(A,B,0)}}$}
\end{picture}
\end{center}
\begin{equation}
\null \label{goffo}
\end{equation}
and it allows for the construction of irreducible representations of
$\mathrm{G_{128}^{(A,B,0)}}$ and all other members of the chain, by
means of the induction algorithm. Such a construction we have not
done, but all the groups of the chain are listed, with their
conjugacy classes in appendix E of \cite{Fre:2015mla}. The group
$\mathrm{G_{128}^{(A,B,0)}}$ leaves the subspace $(A,0,0,B,0,0)$
invariant but still mixes the parameters $A$ and $B$ among
themselves. The subgroup $\mathrm{G_{16}^{(A,B,0)}}\lhd
\mathrm{G_{128}^{(A,B,0)}} $ instead stabilizes the very vector
$(A,0,0,B,0,0)$. This means that any ${\mathrm{(A,B,0)}}$-flow has a
hidden symmetry of order 16 provided by the group
$\mathrm{G_{16}^{(A,B,0)}}$. The general form of these Beltrami
fields is the following one:
\begin{equation}\label{AB0campus}
\mathbf{V}_{(A,B,0)}(\mathbf{r})\, = \,
\mathbf{V}_{(A,B,0)}(x,y,z)\, \equiv \,     \left(
\begin{array}{c}
 2 A \cos (2 \pi  y)+2 B \cos (2 \pi  z) \\
 -2 B \sin (2 \pi  z) \\
 2 A \sin (2 \pi  y) \\
\end{array}
\right)
\end{equation}
In fig.\ref{ABfilotti} we display a plot of the vector field and an
example of equally spaced streamlines.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{ABfildo.png}
\includegraphics[height=80mm]{ABfilini.png}
\end{center}
\caption{\it A plot of the Betrami vector field
$\mathbf{V}_{(A,B,0)}(\mathbf{r})$ (on the left) with $A=5$, $B=7$.
On the right a family of streamlines with equally spaced initial
conditions is displayed.} \label{ABfilotti}
\end{figure}
\par
Looking at  eq.(\ref{goffo}) we notice that there is another group
of order 32, namely  $\mathrm{G_{32}^{(A,A,0)}}$ which contains
$\mathrm{G_{16}^{(A,B,0)}}$ but it is not contained neither in
$\mathrm{G_{128}^{(A,B,0)}} $ nor in $\mathrm{GF_{192}}$. This group
is the stabilizer of the vector $(A,0,0,A,0,0)$ and hence it is the
hidden symmetry group of the flows of type $(A,A,0)$. Once again the
very fact that $\mathrm{G_{32}^{(A,A,0)}}$ is not contained in
$\mathrm{G_{128}^{(A,B,0)}}$ shows that the $(A,A,0)$ flow should
not be considered as a particular case of the $(A,B,0)$-flows rather
as a new type of its own. Let us also stress the difference with the
case of the $(A,A,A)$-flow. Here the hidden symmetry group
$\mathrm{GS_{24}}$ is contained in $\mathrm{GF_{192}}$ and the
interpretation of the $(A,A,A)$-flow as a particular case of the
$(A,B,C)$-flows is permitted. Having set:
\begin{equation}\label{AA0campus}
\mathbf{V}_{(A,A,0)}(\mathbf{r})\, = \,
\mathbf{V}_{(A,A,0)}(x,y,z)\, \equiv \,   A \,  \left(
\begin{array}{c}
 2 A \cos (2 \pi  y)+2 A \cos (2 \pi  z) \\
 -2 A \sin (2 \pi  z) \\
 2 A \sin (2 \pi  y) \\
\end{array}
\right)
\end{equation}
in fig.\ref{AAfilotti} we display a plot of the vector field
$\mathbf{V}_{(A,A,0)}(\mathbf{r})$ and a family of its streamlines.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=80mm]{AAfildostagno.png}
\includegraphics[height=80mm]{AAfili.png}
\end{center}
\caption{\it A plot of the Betrami vector field
$\mathbf{V}_{(A,A,0)}(\mathbf{r})$ (on the left) where are visible
(fat lines) the two stagnation lines of the flow. On the right a
family of streamlines with equally spaced initial conditions is
displayed.} \label{AAfilotti}
\end{figure}
In the case of this flow there are not isolated stagnation points,
rather, because of the $x$-independence of the Beltrami vector
field, there are two entire stagnation lines explicitly given below:
\begin{equation}\label{cannastagnante}
\mathrm{sl}_1\, = \, \left\{x,\frac{1}{2},0\right\}\quad ; \quad
\mathrm{sl}_2\  = \, \left\{x,0,\frac{1}{2}\right\}
\end{equation}
Let us finally come to the case of the flow $(A,0,0)$. The
one-dimensional subspace of vectors of the form $(A,0,0,0,0,0)$ is
left invariant by a rather big subgroup of the classifying group
which is of order $256$. We name it  $\mathrm{G_{256}^{(A,0,0)}}$
and its description is given in appendix E of \cite{Fre:2015mla}. It
is a solvable group with a chain of normal subgroups of index $2$
which ends into a subgroup of order $16$ isomorphic to $\mathbb{Z}_4
\times \mathbb{Z}_4$. This information is summarized in the equation
below:
\begin{center}
\begin{picture}(60,90)
\put(-120,45){$\mathbb{Z}_4 \times \mathbb{Z}_4$}
\put(-75,45){$\sim$} \put (-60,45){$\mathrm{G_{16}^{(A,0,0)}}$} \put
(-20,45){$\vartriangleleft $} \put
(-5,45){$\mathrm{G_{32}^{(A,0,0)}}$} \put (35,45){$\vartriangleleft
$} \put (50,45){$\mathrm{G_{64}^{(A,0,0)}}$} \put
(90,45){$\vartriangleleft $} \put (103,47){\line (1,-1){20}} \put
(126,27){\circle{3}} \put (135,24){$\mathrm{G_{128}^{(A,0,0)}}$}
\put (175,24){$\vartriangleleft $} \put
(190,24){$\mathrm{G_{256}^{(A,0,0)}}$} \put (103,7){\line (1,1){20}}
\put (90,6){$\subset $} \put (-60,6){$\mathrm{G_{16}^{(A,B,0)}}$}
\put (-20,6){$\vartriangleleft $} \put
(-5,6){$\mathrm{G_{32}^{(A,B,0)}}$} \put (35,6){$\vartriangleleft $}
\put (50,6){$\mathrm{G_{64}^{(A,B,0)}}$} \put
(-75,6){$\vartriangleleft $} \put
(-115,6){$\mathrm{G_{8}^{(A,B,0)}}$} \put (-130,6){$\vartriangleleft
$} \put (-170,6){$\mathrm{G_{4}^{(A,B,0)}}$}
\put(-200,6){$\mathbb{Z}_4 $} \put(-185,6){$\sim$}
\end{picture}
\begin{equation}
\label{goffo2} \null
\end{equation}
\end{center}
The group $\mathrm{G_{256}^{(A,0,0)}}$ leaves the subspace
$(A,0,0,0,0,0)$ invariant but occasionally changes the sign of $A$.
The subgroup $\mathrm{G_{128}^{(A,0,0)}} \, \subset \,
\mathrm{G_{256}^{(A,0,0)}}$ stabilizes the very vector
$(A,0,0,0,0,0)$ and therefore it is the hidden symmetry of the
$(A,0,0)$ flows encoded in the planar vector field:
\begin{equation}\label{A00campus}
\mathbf{V}^{(A,0,0)}(\mathbf{r})\, = \,
\mathbf{V}^{(A,0,0)}(x,y,z)\, \equiv \,   A \, \left(
\begin{array}{l}
 \cos (2 \pi  z) \\
 -\sin (2 \pi  z) \\
 0
\end{array}
\right)
\end{equation}
Looking back at equation (\ref{goffo2}) it is important to note that
the group $\mathrm{G_{128}^{(A,0,0)}} \ne
\mathrm{G_{128}^{(A,B,0)}}$ is different from the homologous group
appearing in the group-chain of the $\mathrm{(A,B,0)}$-flows. So
once again the $\mathrm{(A,0,0)}$-flows cannot be regarded as
particular cases of the $\mathrm{(A,B,0)}$-flows. Yet the group
$\mathrm{G_{128}^{(A,0,0)}}$ contains the entire chain of normal
subgroups $\mathrm{G_{128}^{(A,B,0)}}$ starting from
$\mathrm{G_{64}^{(A,B,0)}}$. There is however a very relevant
proviso $\mathrm{G_{64}^{(A,B,0)}}$ is a subgroup of
$\mathrm{G_{128}^{(A,0,0)}}$ but it is not normal.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=50mm]{A00fildus.png}
\includegraphics[height=50mm]{A00filini.png}
\end{center}
\caption{\it A plot of the Beltrami vector field
$\mathbf{V}^{(A,0,0)}(\mathbf{r})$ (on the left). On the right a
family of streamlines with equally spaced initial conditions is
displayed. The planar structure of the streamlines, that are all
straight lines, is quite visible. In the center a standard viewpoint
shot of the streamlines, on the right a view from above. }
\label{A00filucci}
\end{figure}
In fig. \ref{A00filucci} we show a plot of the vector field
$\mathbf{V}^{(A,0,0)}(\mathbf{r})$ and a family of its streamlines.
\section{Temporary Conclusion}
Comparing the group theoretical analysis of the ABC models with
$b$-deformations of the simple considered type it becomes obvious
that there is a link between the symmetry group of a Beltrami-flow
and the surfaces $\Sigma$ that can be utilized to introduce a
$b$-deformed manifold able to host the $b$-deformation of that
Beltrami-flow. At the moment the precise relation between the
boundary surface $\Sigma$ and the symmetry group is by no means
clear yet it is evident that it exists and it should be explored.
Such exploration requires a study of the possible
$\mathfrak{b}$-deformations in the Beltrami flows associated with
higher point group orbits in the momentum lattice. It is obviously a
research direction that should be pursued. Indeed all other Beltrami
flows arising from different instances of the $48$ classes of
momentum vectors have similar structures. The result of the
construction algorithm produces a representation of the Universal
Classifying Group that can be either reducible or irreducible. This
latter can be split into irreps of either $\mathrm{G_{192}}$ or
$\mathrm{GF_{192}}$ and apparently all cases of invariant Beltrami
vector fields have invariance groups that are subgroups of one of
the two groups $\mathrm{G_{192}}$ or $\mathrm{GF_{192}}$. It would
be interesting to transform this observation into a theorem. At the
moment we have not found an obvious proof.
\subsection{A look at the streamlines of the $b$-deformed $AB0$-model} In order to see what the
$b$-deformations might good for, we consider plotting the
$\mathfrak{b}$-deformed field and some of its trajectories. For the
sake of possible applications it is much better to work on compact
spaces rather than on non compact $\mathbb{R}^3$, preserving the
periodicity. As it was already remarked in \cite{cardone2019} as
equation of the boundary, instead of $x=0$, one can choose
$\sin[2\pi x]=0$. So instead of eq.s
(\ref{bigiazione},\ref{pariola}), we get
\begin{equation}\label{sinbigiazione}
    \partial_x \, \longrightarrow \,\sin[2\pi x]\, \partial_x
\end{equation}
and:
\begin{eqnarray}\label{sinpariola}
    \null^{\mathfrak{b}}\mathbf{V}_{ABC}&=&\left(2 A \cos [2 \pi  y]+2 B \cos [2 \pi  z]\right)\,\sin[2\pi x]\, \partial_x \nonumber\\
   &&(2 C \cos [2 \pi  x]-2 B \sin [2 \pi  z])\,\partial_y \,  +\,\left(2 A \sin [2 \pi  y]-2 C
   \sin [2 \pi  x]\right)\,\partial_z
\end{eqnarray}
and all the other formulae in section \ref{bABCsekzia} hold true
upon the substitution of the denominators $1/x$ with $1/\sin[2\pi
x]$. The conclusion remains the same. The $\mathfrak{b}$-deformed
Beltrami equation holds true if and only if $C=0$. In the next
figure \ref{bdefAB0} we present a picture of $\mathfrak{b}$-deformed
$AB0$-field and family of streamlines with the same parameters
$A=5$,$A=7$ utilized for the un-deformed case in fig.
\ref{ABfilotti}.
\begin{figure}[!hbt]
\begin{center}
\includegraphics[height=70mm]{bABmodfildo.png}
\includegraphics[height=70mm]{bABmodelstream.png}
\end{center}
\caption{\it A plot of the Betrami vector field
$\null^{\mathfrak{b}}\mathbf{V}_{ABC}$ (on the left). On the right a
family of streamlines with equally spaced initial conditions is
displayed. What it means to be parallel to the boundary becomes
clear in this picture the trajectories that come very close to $x=0$
or $x=1$ end swinging on the two faces of the cubic cell.  }
\label{bdefAB0}
\end{figure}
\newpage
\chapter{The Landscape Conception with Examples} \label{panoramica} Having clarified the
group theoretical foundations of Arnold--Beltrami Flows we come back
to the issue of producing exact solutions of the Navier Stokes
equations based on velocity fields that satisfy Beltrami equation.
In presence of non vanishing viscosity we can name such solution
\textbf{NS-Beltrami generalized steady flows}
\section{Beltrami equation and generalized steady flows}
The two pillars on which the solutions we consider reside are
provided by the following:
\begin{description}
  \item[A)] Implementation of the \textbf{generalized steady flow} condition  displayed in eq.
  (\ref{genstedona})
  \item[B)] Constancy of the Bernoulli hamiltonian function $H_B$ defined in
  eq.(\ref{finocchionabiscotta})
\end{description}
The pillar B) is easily implemented by setting the pressure field
equal to a constant $h$ minus the squared norm of velocity field:
\begin{equation}\label{tarielka}
    p\left(\mathbf{x},t\right) \, = \, h \, -\, \ft 12 \parallel
    U(\mathbf{x},t)\parallel^2 \, = \,\, h \, - \, \text{const}
    \,\times\,\frac{
    \Omega^{[\mathrm{U}]} \wedge \star_{g} \Omega^{[\mathrm{U}]}}{\text{Vol}}
\end{equation}
where
\begin{equation}\label{volume3forma}
    \text{Vol} \, \equiv \, \ft{1}{3!} \, \mathrm{det}\left(g\right)
    \, \mathrm{d}x\wedge \mathrm{d}y \wedge \mathrm{d}z
\end{equation}
is the volume $3$-form. If the velocity field satisfies Beltrami
equation with eigenvalue $\mu$
\begin{equation}\label{carnenonvale}
    \star_{g} \,\mathrm{d }\Omega^{[\mathrm{U}]} \, = \,  \mu \, \Omega^{[\mathrm{U}]}
\end{equation}
then $\Omega^{[\mathrm{U}]}$ is a \textbf{contact form} and we get:
\begin{equation}\label{pagnuflone}
    \Omega^{[\mathrm{U}]} \wedge \star_{g} \Omega^{[\mathrm{U}]} \,
    = \, \frac{1}{\mu} \, \Omega^{[\mathrm{U}]} \wedge
    \mathrm{d}\Omega^{[\mathrm{U}]} \, = \, \lambda(\mathbf{x},t) \,
    \text{Vol}
\end{equation}
So that the physical pressure field (apart from the additive
constant $h$) obtains an inspiring geometrical interpretation:
indeed it is the nowhere vanishing function $\lambda(\mathbf{x},t)$
mentioned in the definition \ref{ribatriestina} of the Reeb field.
\par
As for pillar A) it is sufficient to recall eq.s
(\ref{laterano},\ref{carnitina}). The essential point is that, as a
consequence of Beltrami equation, the contact one-form
$\Omega^{[\mathrm{U}]}$, whose normalized Reeb field is just the
velocity field $\mathrm{U}\left(\mathbf{x},t\right)$, is an
eigenstate of the Laplace-Beltrami operator $\Delta$ with eigenvalue
$\mu^2$
\begin{equation}\label{kalenkamoya}
    \Delta \,\Omega^{[\mathrm{U}]} \, = \, \mu^2 \, \Omega^{[\mathrm{U}]}
\end{equation}
Then the implementation of the generalized steady flow condition
goes as follows. Consider the finite dimensional vector space
provided by the eigenspace pertaining to the eigenvalue $\mu$:
\begin{equation}\label{ramazzato}
 \mathbf{V}_\mu \ni \Omega^{[\mathrm{u}]} \quad \Rightarrow \quad   \star_{g} \,\mathrm{d }\Omega^{[\mathrm{u}]} \, = \,  \mu \,
    \Omega^{[\mathrm{u}]} \quad ; \quad \Omega^{[\mathrm{u}]} \, =
    \, \sum_{i=1}^{N_\mu} \, F_i \, \Omega^{[\mathrm{u}_i]}
\end{equation}
where $\mathrm{u}_i(\mathbf{x})$ are the normalized Reeb fields of a
basis of solutions $\Omega^{[\mathrm{u}_i]}$ and $F_i$ the free
parameters spanning the eigenspace $\mathbf{V}_\mu$. The number
$N_\mu$ is the degeneracy of the eigenvalue $\mu$ namely the
dimension of the eigenspace.  Next subdivide the $\mathbf{V}_\mu$ in
two freely chosen subspaces:
\begin{equation}\label{rominaEalbano}
    \mathbf{V}_\mu \, = \, \mathbf{V}_\mu^0 \oplus \mathbf{V}_\mu^t
    \quad ; \quad \mathrm{dim}\,\mathbf{V}_\mu^0\, = \, M_0 \, < \, N_\mu \quad ;
    \quad \mathrm{dim}\,\mathbf{V}_\mu^t \, = \, N_\mu\, -\, M_0
\end{equation}
Correspondingly the contact form $\Omega^{[\mathrm{u}]}$ and its
normalized Reeb field $\mathrm{u}(\mathbf{x})$ will split in two
parts:
\begin{equation}\label{strappone}
    \Omega^{[\mathrm{u}]} \, = \, \Omega^{[\mathrm{u}^0]} \, + \,
    \Omega^{[\mathrm{u}^t]} \quad ; \quad \Omega^{[\mathrm{u}^0]}
    \in \mathbf{V}_\mu^0 \quad ; \quad \quad \Omega^{[\mathrm{u}^t]}
    \in \mathbf{V}_\mu^t
\end{equation}
Then setting the driving force as follows:
\begin{equation}\label{patentediguida}
    \mathbf{f}\, = \, -\, \nu \,\mu \,\Omega^{[\mathrm{u}^0]}
\end{equation}
and the contact form (Reeb field) as follows
\begin{equation}\label{NStanda}
    \Omega^{[\mathrm{U}]}\, = \, \Omega^{[\mathrm{u}^0]}+
    \exp\left[- \mu^2 \,t\right]\,\Omega^{[\mathrm{u}^t]}
\end{equation}
the generalized steady flow condition (\ref{genstedona}) is
satisfied and the velocity field
\begin{equation}\label{corricorri}
    \mathrm{U}(\mathbf{x},t) \, = \, \mathrm{u}^0(\mathbf{x}) \, +
    \,\exp\left[- \mu^2 \,t\right] \,\mathrm{u}^t(\mathbf{x})
\end{equation}
fulfils the Navier Stokes equation (\ref{EulerusEqua}).
\section{The landscape conception}
It follows from the above discussion that the main issue in order to
construct the NS-Beltrami generalized steady flows is the
construction of the eigenspaces $\mathbf{V}_\mu$ and their
organization in subspaces according with symmetry principles. This
is what leads to the \textit{landscape conception}.
\par
When the manifold $\mathcal{M}_3$ is the torus defined by
eq.(\ref{metricT3}), the construction of the eigenspace
$\mathbf{V}_\mu$ can be performed geometrically, relying on the
algorithm explained in section \ref{algoritmo} and on the orbits of
the point group $\mathfrak{P}_\Lambda$ in the momentum lattice
$\Lambda^\star$. We just need to consider all those orbits for which
the squared norm of the momentum vectors $\mathbf{k}$ is the same.
Geometrically this amounts to consider the spherical layers of
radius $r = \sqrt{\mathbf{k}^2}$ defined as the intersection of a
sphere of such a radius with the momentum lattice:
\begin{equation}\label{spherlayer}
    \mathrm{SL}_{\mathbf{k}^2}\, \equiv  \,
    \mathbb{S}_{r=\sqrt{\mathbf{k}^2}}\, \bigcap \, \Lambda \quad ;
    \quad \mathbf{k} \, \in \, \Lambda
\end{equation}
In each spherical layer $\mathrm{SL}_{\mathbf{k}^2}$ we find a
certain finite number of points which in the average steadily
increases with $\mathbf{k}^2$ and provides a fluctuating number of
point group orbits which, however, also steadily increases with
$\mathbf{k}^2$. Hence to each layer we can associate a solution of
Beltrami equation whose degeneray $N_{\mathbf{k}^2}$ is a finite
number in the following range:
\begin{equation}\label{laborrino}
|\mathrm{SL}_{\mathbf{k}^2}|\,  \leq N_{\mathbf{k}^2}\, \leq\,  2
|\mathrm{SL}_{\mathbf{k}^2}|   \quad ; \quad
|\mathrm{SL}_{\mathbf{k}^2}| \, \equiv \, \#\text{ of points in
    $\mathrm{SL}_{\mathbf{k}^2}$}
\end{equation}
This solution of Beltrami equation constitutes a reducible
representation of the Universal Classifying Group
$\mathfrak{UG}_\Lambda$ of dimension $N_{\mathbf{k}^2}$
\begin{equation}\label{ramirezsancho}
    \mathrm{SL}_{\mathbf{k}^2} \, \stackrel{\text{Beltrami
    field}}{\Longrightarrow} \,
    D\left[\mathfrak{UG}_\Lambda,N_{\mathbf{k}^2}\right]
\end{equation}
which can be decomposed into irreps
\begin{equation}\label{sferodecompo}
    D\left[\mathfrak{UG}_\Lambda,N_{\mathbf{k}^2}\right] \, = \,
    \bigoplus_{i=1}^{\mathfrak{r}}\, a_i \,
    \,D_i\left[\mathfrak{UG}_\Lambda,n_i\right] \quad ; \quad
    \sum_{i}^{\mathfrak{r}} \,a_i \, n_i \, = \, N_{\mathbf{k}^2}
\end{equation}
having denoted by $\mathfrak{r}$ the number of conjugacy classes and
hence of irreps of $\mathfrak{UG}_\Lambda$, by $n_i$ the dimension
of the $i$-th irrep and by $a_i$ its multiplicity. Since
$N_{\mathbf{k}^2} \to \infty$ when $\mathbf{k}^2 \to \infty$ it is
obvious that enlarging the landscape the same representations will
reappear again and again with increasing multiplicity.
\par
The two MATHEMATICA codes described in chapter \ref{descrivomath}
are finalized to:
\begin{description}
  \item[a)] to the construction of a large landscape
  \item[b)] to the
construction of the Beltrami solution on each chosen spherical layer
of that landscape
  \item[c)] to the group theoretical analysis of the corresponding
representation
$D\left[\mathfrak{UG}_\Lambda,N_{\mathbf{k}^2}\right]$ including its
further decomposition with respect to subgroups of
$\mathfrak{UG}_\Lambda$.
\end{description}
\section{Sketches of the cubic and hexagonal landscapes}
In this section we flash through a pair of inspiring examples from
both instances of main lattice families, the cubic and hexagonal
ones.
\subsection{The cubic landscape}
Utilizing the background MATHEMATICA code
\textbf{UniClasGroupCubicLat}, in the NoteBook
\textbf{CubicLandscape} we have constructed a rather large portion
of the self-dual cubic lattice $\Lambda_{cubic}$ containing 117649
lattice points. In this portion of the lattice we found 1057
spherical layers that we  analyzed with our computer code. In this
way we found a maximally large representation of dimension 792
residing on the largest radius sphere hosted by this lattice region:
\begin{equation}\label{massimone}
     \text{Max} \text{Dim}\, = \, \dim  D\left[
G_{1536}, 792\right]\, = \, 792 \quad \Leftrightarrow\quad
\left|\mathbf{k}\right|^2 =689
\end{equation}
The distribution number of points and dimensions is displayed in
fig.\ref{distribuocubo}
\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=100mm]{panoramacubico.png}
\end{center}
\caption{\it A view of the distribution of $\#$ of points in
spherical layer as $\mathbf{k}^2 \to \infty$. The maximum for a
finite portion of the lattice is not typically for the highest
radius, since as the radius grows the lattice points are more
scattered on the spherical surface, yet if we continue to enlarge
the sphere it will intercept more an more lattice points and reach
new higher maxima after some oscillations. } \label{distribuocubo}
\end{figure}
\newpage
\subsection{An example of Chaos from symmetry from the cubic
lattice} As an illustration of the Beltrami construction we
considered the Beltrami fields associated with a specific layer
namely that one where:
\begin{equation}\label{layer}
    |\mathbf{k}|^2 = 576
\end{equation}
We find that the number of points on this layer is 30 that arrange
themselves in a point group orbit $\mathcal{O}_6$ of length 6  plus
another one $\mathcal{O}_{24}$ of length 24. The Beltrami solution
corresponding to this layer has therefore eigenvalue $\mu \, = \, 24
\pi$ and the reducible representation of the Universal Classifying
Group is found to decompose into irreps as follows:
\begin{equation}\label{singolodecompo}
    D\left[G_{1536},30\right] \, = \, D_1\left[G_{1536},1\right]+D_2\left[G_{1536},1\right]+2 D_5\left[G_{1536},2\right]
    +4 D_7\left[G_{1536},3\right]+4
   D_8\left[G_{1536},3\right]
\end{equation}
As one sees the considered layer contains one singlet of the maximal
possible symmetry group. It is interesting to visualize both the
plot of this vector field and some of its trajectories. In
fig.\ref{vettoronesing}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=80mm]{singolovectplotcub.png}
\includegraphics[width=80mm]{singcubsliced.png}
\end{center}
\caption{\it In this figure we show the three dimensional vector
plot of the unique Beltrami vector field invariant under the largest
symmetry group $G_{1536}$ that arises in the eigenspace pertaining
to the Beltrami eigenvalue $\mu \, = \, 24$ namely on the spherical
layer $\mathbf{k}^2 \, = \, 576$. The high symmetry of the vector
field is almost evident at eye-sight. } \label{vettoronesing}
\end{figure}
Next we show the example of just one trajectory and of 27 equally
spaced streamlines of this symmetric vector field that we have
followed for 50 iterations of numerical integrations. The plots are
displayed in fig.\ref{gomitolinocub}
\newpage
\begin{figure}[!hb]
\begin{center}
\includegraphics[width=70mm]{unatraggiacubica.png}
\includegraphics[width=70mm]{gattinocubico.png}
\end{center}
\caption{\it In the picture on the left we show just one fluid
element trajectory starting in a randomly chosen point
$p=\{1/7,2/9,5/33\}$. In the picture on the right we display the
plot of 27 streamlines whose starting point are equally spaced over
the three dimensions. After 50 integration cycles they make an
inextricable pattern. This is the visual manifestation of the
contact structure. \label{gomitolinocub}}
\end{figure}
\subsection{The hexagonal landscape}
As for the hexagonal lattice we have so far constructed a landscape
portion portion of the infinite momentum lattice that is shaped as a
polyhedron with an hexagonal basis and it is displayed
fig.\ref{bottigliaesagonale}. This landscape contains 33084
\textit{interior points} and 3888 \textit{points on its boundary}.
This distinction has no intrinsic meaning and it simply corresponds
to the geometrical shape of the considered lattice portion.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=70mm]{momentbig3d.png}
\includegraphics[width=70mm]{momentbigplanar.png}
\end{center}
\caption{\it The portion of considered hexagonal momentum lattice is
taken to be a polyhedron with an hexagonal basis that is shown on
the right, extended in the $z$-direction just as much as it extends
in the $xy$-plane. The lattice points on the 6 lateral faces of this
polyhedron have been displayed in green while the lattice  points
that are inside the polyhedron have been displayed in brown. There
are 3888 points inside the polyhedron and 4800 points on the 6
faces. Note that for visual convenience we have aligned the $z$-axis
horizontally and the $y$-axis vertically.
\label{bottigliaesagonale}}
\end{figure}
We have intersected this polyhedron shaped portion of the lattice
with spheres and we have found 544 spherical layers with a
distribution of number of points as shown in
fig.\ref{distribuoesagono}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=120mm]{panoramaesagonale.png}
\end{center}
\caption{\it Distribution of the $\#$ of points on the spherical
layers of the hexagonal moment lattice. \label{distribuoesagono}}
\end{figure}
This result was obtained utilizing the background MATHEMATICA
Notebook \textbf{HexagBckgroundN6.nb} described in appendix
\ref{curucu}. After initialization obtained by typing:
\begin{verbatim}
initiomute
\end{verbatim}
we obtained the 544 records corresponding to each layer, organized
in decreasing order with respect to the number of contained point by
giving the command:
\begin{verbatim}
preparostratatubemute
\end{verbatim}
and inserting in the dialogue box prompted by computer the datum:
\begin{verbatim}
nplan = 18
\end{verbatim}
which establishes the number of points on each bone of the skeleton
($2\,\text{nplan}+1$). The set of records is depositated in a file
named \textbf{reorgfiltro}.
\section{An example of chaos from
symmetry in the hexagonal landscape}
Among the records of this
landscape we have considered the spherical layer defined by:
\begin{equation}\label{ciusone}
    \mathbf{k}^2 \, = \, \frac{128}{3}
\end{equation}
and catalogued as $Id=132$  (\textit{i.e.} in
\textbf{reorgfiltro[[132]]}) which contains 90 lattice point. These
90 lattice points intercepted by the sphere of radius
$\sqrt{\frac{128}{3}}$ are organized in the following orbits of the
point group $\mathrm{Dih_6}$:
\begin{eqnarray}\label{strato132}
\mathbb{S}_{r^2=\frac{128}{3}}\bigcap\Lambda^\star_{hexag} & = &
O_1(\{6,1\})+O_2(\{12,3\})+O_3(\{12,2\})+O_4(\{12,2\})\nonumber\\
&&+O_5(\{12,2\})+O_6(\{12,2\}) +O_7(\{12,2\})+O_8(\{12,2\})
\end{eqnarray}
and yield a $90 \times 90 $-dimensional representation of the
Universal Classifying Group $\mathfrak{U}_{72}$ which admits the
following decomposition into irreps:
\begin{eqnarray}
\label{novantina} D\left[\mathfrak{U}_{72},90\right] &=&  2
D_1(\mathfrak{U}_{72},1)+2 D_3(\mathfrak{U}_{72},1)+3
D_5(\mathfrak{U}_{72},1)+3 D_7(\mathfrak{U}_{72},1)+5
   D_{10}(\mathfrak{U}_{72},2)\nonumber\\
   &&+5 D_{11}(\mathfrak{U}_{72},2)+5 D_{13}(\mathfrak{U}_{72},2)+5
   D_{15}(\mathfrak{U}_{72},2)+5 D_{17}(\mathfrak{U}_{72},2)\nonumber\\
   &&+5 D_{19}(\mathfrak{U}_{72},2)+5
   D_{21}(\mathfrak{U}_{72},2)+5 D_{24}(\mathfrak{U}_{72},2)
\end{eqnarray}
This result is obtained by typing the command
\begin{verbatim}
solubo
\end{verbatim}
choosing the layer $132$ in the dialogue box and then answering $No$
when the question is posed whether you want to truncate the analysis
to a single orbit or extend it to the entire layer.
\par
As one sees from eq.(\ref{novantina}) the 90-dimensional parameter
space contains a $2$-dimensional subspace invariant with respect to
the full group $\mathfrak{U}_{72}$, corresponding to the identity
representation. It is interesting to choose such an example and
consider its properties.
\subsection{Choice of the $\mathfrak{U}_{72}$ invariant subspace}
The restriction of the Beltrami 90 parameter field to singlet
subspace can be operated by means of the subroutine
\textbf{NSeqsolvetrad}. Typing the command
\begin{verbatim}
NSeqsolvetrad
\end{verbatim}
The computer extracts from the result of the previous calculation
the 90-parameters vector field and renames it \textbf{VectField};
then it verifies that  indeed it satisfies Beltrami equation. Then
the computer remakes, in a more extended way, the previously done
group-theoretical analysis including also the construction of  the
projectors on the irreducible subspaces. It informs you about the
list of $12$ irreps:
\begin{equation}\label{labinocola}
 \textit{contributing irreps} \, = \,  \{1,3,5,7,10,11,13,15,17,19,21,24\}
\end{equation}
that are contributing to the 90-dimensional reducible representation
and eventually it performs a change of basis in parameter space
where the new parameters $Y_{i,j}$ have two indices, the first
indicating the \textit{irrep}, the second running on as many values
as the dimension of the irrep times its multiplicity. The final
outcomes of the routine \textbf{NSeqsolvetrad} are the following:
\begin{enumerate}
  \item The array \textbf{VFF} that contains the entire Beltrami
  field with reorganized coefficients
  \item The array \textbf{redpara} that contains the list of
  redefined 90 parameters
\end{enumerate}
The computer stops and invites you to type:
\begin{equation*}
\text{You type \textbf{vivi} = } \{Y_{p_1,q_1},Y_{p_2,q_2},\dots\}
\, = \, \text{a set of $Y_{p,q}$}
\end{equation*}
The set \textbf{vivi} will be preserved while all the other
$Y_{p,q}$ will be set to zero in a new version of the Beltrami
field. This allows you to select the irreps you desire. Since we are
interested in the case of singlet  we set:
\begin{equation*}
\text{\textbf{vivi} = } \{Y_{1,1},Y_{1,2}\}
\end{equation*}
and we type the command
\begin{verbatim}
selectresume
\end{verbatim}
The computer makes the choice and tells you that the desired
Beltrami vector field is encoded in the array \textbf{Vasym}. Later,
should you  desire to make a different choice, the original complete
result is preserved in \textbf{VFF} and you just have to type the
command \textbf{selectus}, type another set of \textbf{vivi} and
conclude the operation by typing \textbf{selectresume}.
\par
For the moment we stick to the singlet case.
\subsection{Analysis of the singlet Beltrami vector field}
\label{singlechaos} Collecting respectively the coefficient of
$Y_{1,1}$ and $Y_{1,2}$ and using the hexagonal cell coordinates
$u,v,r$ defined by:
\begin{equation}\label{uvrcoordi}
    x\, = \,\frac{2 u-v}{\sqrt{2}} \quad ; \quad y \, = \,\sqrt{\frac{3}{2}} v\quad ;
    \quad z\, = \, \sqrt{2} \, r
\end{equation}
we obtain two explicit vector fields
$\mathbf{V}^{sing|1,2}\left(u,v,r\right)$ of which, due to the
massiveness of the formulae, we display only the first, in order to
give the reader some feeling of the result structure and quality.
Here it is:
\begin{eqnarray}
\label{sing1c1}
  \mathbf{V}^{sing|1}_1 &=& \frac{1}{8
   \sqrt{3}}\left\{11 \sin (2 \pi  (6 r+3 u-7 v))-28 \sin (2 \pi  (6 r+4 u-7 v))-17 \sin (2 \pi  (6
   r+7 u-4 v))\right.\nonumber\\
   &&\left.-17 \sin (2 \pi  (6 r+7 u-3 v))+17 \sin (2 \pi  (6 r-7 u+3 v))-28 \sin (2
   \pi  (-6 r+4 u+3 v))\right.\nonumber\\&&\left.-28 \sin (2 \pi  (6 r+4 u+3 v))+17 \sin (2 \pi  (6 r-7 u+4 v))  +11
   \sin (2 \pi  (-6 r+3 u+4 v))\right.\nonumber\\
   &&\left.+11 \sin (2 \pi  (6 r+3 u+4 v))+28 \sin (2 \pi  (6 r-4
   u+7 v))-11 \sin (2 \pi  (6 r-3 u+7 v))\right.\nonumber\\
   &&\left.-32 \cos (2 \pi  (6 r+3 u-7 v))-16 \cos (2 \pi
   (6 r+4 u-7 v))-16 \cos (2 \pi  (6 r+7 u-4 v))\right. \nonumber\\
   && \left.+16 \cos (2 \pi  (6 r+7 u-3 v))-16 \cos
   (2 \pi  (6 r-7 u+3 v))-16 \cos (2 \pi  (-6 r+4 u+3 v))\right.\nonumber\\
   &&\left. +16 \cos (2 \pi  (6 r+4 u+3
   v))+16 \cos (2 \pi  (6 r-7 u+4 v))-32 \cos (2 \pi  (-6 r+3 u+4 v))\right.\nonumber\\
   &&\left.+32 \cos (2 \pi  (6
   r+3 u+4 v))+16 \cos (2 \pi  (6 r-4 u+7 v))+32 \cos (2 \pi  (6 r-3 u+7 v))\right\}\nonumber\\
\end{eqnarray}
\begin{eqnarray}
\label{sing1c2} \mathbf{V}^{sing|1}_2 &=& \frac{1}{8} \left( 15 \sin
(2 \pi (6 r+3 u-7 v))+2 \sin (2 \pi  (6 r+4 u-7 v))+13 \sin (2 \pi
    (6 r+7 u-4 v))\right.\nonumber\\
&&\left.  -13 \sin (2 \pi  (6 r+7 u-3 v))+13 \sin (2 \pi  (6 r-7 u+3
v))-2 \sin
   (2 \pi  (-6 r+4 u+3 v))\right.\nonumber\\
   &&\left.-2 \sin (2 \pi  (6 r+4 u+3 v))-13 \sin (2 \pi  (6 r-7 u+4
   v)) \right.\nonumber\\
&&\left.  -15 \sin (2 \pi  (-6 r+3 u+4 v))-15 \sin (2 \pi  (6 r+3
u+4 v))-2 \sin (2 \pi  (6
   r-4 u+7 v)) \right.\nonumber\\
&&\left.  -15 \sin (2 \pi  (6 r-3 u+7 v))-16 \cos (2 \pi  (6 r+4 u-7
v))-16 \cos (2
   \pi  (6 r+7 u-4 v)) \right.\nonumber\\
&&\left.  -16 \cos (2 \pi  (6 r+7 u-3 v))+16 \cos (2 \pi  (6 r-7 u+3
v))+16
   \cos (2 \pi  (-6 r+4 u+3 v)) \right.\nonumber\\
&&\left.  -16 \cos (2 \pi  (6 r+4 u+3 v))+16 \cos (2 \pi  (6 r-7
   u+4 v))+16 \cos (2 \pi  (6 r-4 u+7 v)) \right.\nonumber\\
\end{eqnarray}
\begin{eqnarray}
\label{sing1c3} \mathbf{V}^{sing|1}_3 &=& \frac{1}{4
\sqrt{3}}\,\left\{11 \sin (2 \pi (6 r+3 u-7 v))+11 \sin (2 \pi  (6
r+4 u-7 v))+11 \sin (2 \pi  (6
   r+7 u-4 v)) \right.\nonumber\\
&&\left.  +11 \sin (2 \pi  (6 r+7 u-3 v))+11 \sin (2 \pi  (6 r-7 u+3
v))-11 \sin (2
   \pi  (-6 r+4 u+3 v))\right.\nonumber\\
   &&\left.+11 \sin (2 \pi  (6 r+4 u+3 v))-8 \cos (2 \pi  (6 r-4 u+7 v))+8 \cos (2 \pi  (6 r-3 u+7 v)) \right.\nonumber\\
&&\left.  +11 \sin (2 \pi  (6 r-7 u+4 v))-11
   \sin (2 \pi  (-6 r+3 u+4 v))+11 \sin (2 \pi  (6 r+3 u+4 v)) \right.\nonumber\\
&&\left.  +11 \sin (2 \pi  (6 r-4
   u+7 v))+11 \sin (2 \pi  (6 r-3 u+7 v))+8 \cos (2 \pi  (6 r+3 u-7 v))\right.\nonumber\\
   &&\left. -8 \cos (2 \pi
   (6 r+4 u-7 v)) +8 \cos (2 \pi  (-6 r+4 u+3 v))+8 \cos (2 \pi  (6 r+4 u+3 v)) \right.\nonumber\\
&&\left.  +8 \cos (2 \pi  (6 r+7 u-4 v))-8 \cos (2 \pi  (6 r+7 u-3
v))-8 \cos (2
   \pi  (6 r-7 u+3 v)) \right.\nonumber\\
&&\left.  +8
   \cos (2 \pi  (6 r-7 u+4 v))-8 \cos (2 \pi  (-6 r+3 u+4 v))-8 \cos (2 \pi  (6 r+3 u+4
   v)) \right\}\nonumber\\
\end{eqnarray}
In order to perceive what \textit{chaos from symmetry} really means
we focus on the above singlet Beltrami vector field  and we make a
vector plot of it inside the cubic shaped fundamental cell
$u\in[0,1],v\in[0,1],r\in[0,1]$ which can be smoothly mapped into
one of the three sectors of the hexagonal cell.
\par
The procedure is the following. First you type the command
\begin{verbatim}
mainvisualizehexag
\end{verbatim}
This initializes all the necessary items for the visualization of
vector fields in the hexagonal cell. The computer stops and tells
you what follows:
\begin{verbatim}
Next you have to choose the analytic vector field you want to study
setting VKV = ?
\end{verbatim}
The vector field should be provided as a function of the hexagonal
coordinates $u,v,r$ although in the standard orthogonal basis.
Furthermore the computer asks you to write
\begin{verbatim}
suballo = {..->?,..->?}
\end{verbatim}
This is a substitution rule that gives explicit numerical values to
the parameters contained in \textbf{VKV} if there are any.
\par
After you typed such an information you can type the command:
\begin{verbatim}
preparsolhexag
\end{verbatim}
and you obtain as a result the vector plot of your vector field
together with the instructions in order to proceed to the
calculation streamlines by means of numerical integration of the
first order differential equations (\ref{strimotti}):
\begin{verbatim}
Next you have to choose whether you  want to see a single trajectory
or a family of streamlines. For a single trajectory you have to type
the command singletracthexag, otherwise you type the command
streamlineshexag
\end{verbatim}
The result for our singlet field is displayed in
fig.\ref{ordineesagonale}.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=100mm]{ordinesag.png}
\end{center}
\caption{\it Plot of the vector field displayed in eq.s
(\ref{sing1c1}-\ref{sing1c3}) which is invariant with respect to the
full group $\mathfrak{U}_{72}$. The high symmetry of this vector
field is visible at eight sight. The pattern repeats itself under
rotation and reflections but also under the $1/6$ translations in
the vertical direction $r$. \label{ordineesagonale}}
\end{figure}
\par
Given the high symmetry of the vector plot the capricious chaotic
development of the stream-lines follows from the integration of the
first order equations. As an exemplification we begin with a single
stream line starting at a generic initial point of the hexagonal
fundamental cell.
\par
We choose:
\begin{equation}\label{inicondahex}
    \{u_0,v_0,r_0\} \, = \, \left\{\frac{2}{7},\frac{4}{9},\frac{2}{15} \right\}
\end{equation}
that we supply in the dialogue box which is prompted by the computer
upon typing the command:
\begin{verbatim}
singletracthexag
\end{verbatim}
The response elaborated by the computer is the wandering path
presented in fig.\ref{vadoaspassoesag}.
\begin{figure}[!htb]
\begin{flushleft}
\includegraphics[width=60mm]{singoltrachex.png}
\includegraphics[width=50mm]{usingplot.png}
\includegraphics[width=50mm]{vsingplot.png}
\includegraphics[width=50mm]{rsingplot.png}
\end{flushleft}
\caption{\it Plot of the single streamline starting at the point
(\ref{inicondahex})  for the vector field invariant under
$\mathfrak{U}_{72}$ defined in eq.s (\ref{sing1c1}-\ref{sing1c3})
and displayed in  fig. \ref{ordineesagonale}. In the first picture
we present the three-dimensional path of the fluid element within
the hexagonal cell, whose basis is divided in the three sectors,
related to each other by a $2\pi/3$ rotation. In green we have the
fundamental sector, image of the square $u\in[0,1]$,$v\in[0,1]$. The
other pictures display the time plots $u(t),v(t),r(t)$ of the three
hexagonal coordinates. The chaotic behavior is fully evident.
\label{vadoaspassoesag}}
\end{figure}
\par
Next we proceed to the calculation of 25 streamlines that have
equally spaced starting points in the fundamental planar cell
$u\in[0,1]$, $v\in[0,1]$ but after $30$ integration steps have
already every diffused capriciously and chaotically throughout the
entire hexagonal cell.
\par
To this command you type the command:
\begin{verbatim}
streamlineshexag
\end{verbatim}
You will be prompted a few questions to be answered in dialogue
boxes. The first question concerns the value \textbf{nph} which
should be an integer number. Once this value is fixed, the computer
will consider the $u_0,v_0$ coordinates of the streamline starting
points as being located at:
\begin{equation}\label{spaziolino}
    \{u_0,v_0\}\, = \, \left\{
    \frac{p}{\mathbf{nph}},\frac{q}{\mathbf{nph}}\right\} \quad ;
    \quad \left \{ \begin{array}{ccc}
                     p & = & 0,1,\dots,\mathbf{nph} \\
                     q & = & 0,1,\dots,\mathbf{nph}
                   \end{array}\right.
\end{equation}
Secondly the computer will pose the question:
\begin{verbatim}
Do you want a vertical grid or only the base, Yes or No?
\end{verbatim}
If you choose $No$, the starting point of the streamlines will be
only on the base of the hexagonal cell in the first of the three
fundamental sectors that are rotated one into the other by $2\pi/3$
rotations. So you will get $\textbf{nplan+1}^2$ streamlines. If you
answer $Yes$ you will be prompted another dialogue box where you
have to specify the value \textbf{npv} which must be a positive
integer as well. In this case the grid of initial points for the
streamlines will be three dimensional as follows:
\begin{equation}\label{spazzolone}
    \{u_0,v_0,r_0\}\, = \, \left\{
    \frac{p}{\mathbf{nph}},\frac{q}{\mathbf{nph}},\frac{s}{\mathbf{npv}}\right\} \quad ;
    \quad \left \{ \begin{array}{ccc}
                     p & = & 0,1,\dots,\mathbf{nph} \\
                     q & = & 0,1,\dots,\mathbf{nph}\\
                     s & = & 0,1,\dots,\mathbf{npv}
                   \end{array}\right.
\end{equation}
In our case we choose $\mathbf{nph}\,=\,4$ and we restrict ourselves
to streamlines originating in the first fundamental sector of the
hexagonal basis. The result is what you see in figure
\ref{casinonellesagono}.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=100mm]{uncasinoesag.png}
\end{center}
\caption{\it Plot of 25 streamlines, all originating at equally
spaced points (\ref{spaziolino}) in the fundamental sector (green
parallelogram) of the hexagonal basis  for the vector field
invariant under $\mathfrak{U}_{72}$ defined in eq.s
(\ref{sing1c1}-\ref{sing1c3}) and displayed in  fig.
\ref{ordineesagonale}. The chaotic behavior after $t_{max}=20$ units
of integration time are quite evident.\label{casinonellesagono}}
\end{figure}
\section{A vertical motion} The main problem one meets in several
applications of hydrodynamics is, as we already stressed, that of
mixing a chaotic behavior at small scales with an approximate global
motion, at larger scales, in one definite direction that we can
conventionally assume to be the $z$-axis. The superposition is
intrinsically forbidden by the non linearity of the NS and Euler
equations, yet within the scope of the Beltrami fields and the
landscape approach there is a limited superposition freedom:
\textit{Beltrami flows having the same eigenvalue parameter
$\lambda$ can be linearly combined.} Hence it is interesting to
consider whether in the same spherical layer that contains highly
symmetric and hence chaotic flows like that described in the
previous section \ref{singlechaos} there are other orbits that
provide instead rather orderly flows uniformly directed. The answer
is yes and it is also of a general type. All orbits of the point
group $\mathrm{Dih_6}$ in the momentum lattice
$\Lambda_{exag}^\star$ that are of type $O({6,1})$ have the
following features:
\begin{description}
\item[a)] The orbits is planar at $z=0$
\item[b)] The Beltrami flow associated with the orbit has a 6 dimensional parameter space that decomposes with respect
to the $\mathfrak{U}_{72}$ group according to the following scheme:
\begin{equation}\label{localinosfizioso}
    D[\mathfrak{U}_{72},6]\, = \,
    D_\alpha[\mathfrak{U}_{72},1]+D_\beta[\mathfrak{U}_{72},1]+D_\gamma[\mathfrak{U}_{72},2]+D_\delta[\mathfrak{U}_{72},2]
\end{equation}
where $D_\alpha,D_\beta$ are two different one-dimensional and
$D_\gamma,D_\delta$ are two different two-dimensional
representations.
\item[c)] The restriction of the Beltrami field to the two
one-dimensional representations provides an integral model whose
streamlines are parallel spirals directed in the $z$ direction that
wind around their central vertical axis with wider or more tight
coils.
\end{description}
In the case of the $O({6,1})$ orbit contained in the decomposition
(\ref{strato132}) of the spherical layer $Id=132$ we obtain
\begin{equation}\label{larino}
    D[\mathfrak{U}_{72},6]\, = \,D_5(\mathfrak{U}_{72},1)+D_7(\mathfrak{U}_{72},1)
    +D_{11}(\mathfrak{U}_{72},2)+D_{17}(\mathfrak{U}_{72},2)
\end{equation}
and restricting the parameter space to the two one-dimensional
representations we obtain the following Beltrami field already
rewritten in terms of the hexagonal coordinates:
\begin{equation}\label{cortemilia}
    \mathbf{V}^{vert}\, = \, \alpha \,\mathbf{V}^{vert}_1 \, + \, \beta
    \,\mathbf{V}^{vert}_2
\end{equation}
where
\begin{equation}\label{curtu1}
    \mathbf{V}^{vert}_1 \, = \, \left(
\begin{array}{l}
 \frac{1}{9} \left(\sin [16 \pi  (u-v)]-\sin [16 \pi  u]-2 \sin [16 \pi  v]\right) \\
 \frac{1}{3 \sqrt{3}}\left(\sin [16 \pi  (u-v)]+\sin [16 \pi  u])\right) \\
 \frac{1}{9} \left(2 \cos [16 \pi  (u-v)]+2 \cos [16 \pi  u]+2 \cos [16 \pi  v]\right) \\
\end{array}
\right)
\end{equation}
and
\begin{equation}\label{curtu2}
    \mathbf{V}^{vert}_2 \, = \,\left(
\begin{array}{l}
 \frac{1}{9} \left(2 \cos [16 \pi  (u-v)]+2 \cos [16 \pi  u]-\cos [16 \pi  v]\right) \\
 \frac{1}{3 \sqrt{3}}\left(2 \cos [16 \pi  (u-v)]-2 \cos [16 \pi  u]\right) \\
 \frac{1}{9} (-4 \sin [16 \pi  (u-v)])+4 \sin [16 \pi  u]-\sin [16 \pi  v]) \\
\end{array}
\right)
\end{equation}
We consider the behavior of the vector field:
\begin{equation}\label{condrino}
    \mathbf{V}^{vert|11}\,\equiv \,\mathbf{V}^{vert}_1 \, + \, \mathbf{V}^{vert}_2
\end{equation}
utilizing the same tokens as above and we obtain the vector plot
displayed in fig.\ref{vertone} together with the display of the
central streamline departing from the origin $\{0,0,0\}$.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=70mm]{vertaliot.png}
\includegraphics[width=70mm]{cobra.png}
\end{center}
\caption{\it Plot of the vector field defined in eq.(\ref{condrino})
sided by the 3D plot of its central stream line which is a vertical
climbing up spiral. \label{vertone}}
\end{figure}
\newpage
In fig.\ref{cobratwo} we show another spiral streamline  climbing up
vertically from another interior point of the fundamental planar
cell.
\begin{figure}[!htb]
\begin{flushleft}
\includegraphics[width=50mm]{cobra2.png}
\includegraphics[width=50mm]{uplotcob.png}
\includegraphics[width=50mm]{vplotcob.png}
\includegraphics[width=50mm]{rplotcob.png}
\end{flushleft}
\caption{\it Plot of a single streamline starting at the point
$\{\ft 37, \ft 13, 0\}$ for the vector field defined in eq.s
(\ref{curtu1}-\ref{condrino}) and displayed in  fig. \ref{vertone}.
 \label{cobratwo}}
\end{figure}
Depending from the position in the fundamental cell, the spiral can
be directed upwards or downwards. In fig. \ref{girinispaventati} we
display the plot of another instance of the vertical Beltrami field,
namely the following one:
\begin{equation}\label{papaghena}
    \mathbf{V}^{vert|1,-2}\,\equiv \,\mathbf{V}^{vert}_1 \, - \,2\, \mathbf{V}^{vert}_2
\end{equation}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=80mm]{1m2fildo.png}
\includegraphics[width=80mm]{mollematras.png}
\end{center}
\caption{\it Plot of the Beltrami field in eq.\ref{papaghena} sided
by a plot of 25 of its streamlines, originating from the fundamental
planar cell. As one realizes some are uprising spirals, some other
descending spirals. \label{girinispaventati}}
\end{figure}
\section{Combining vertical motion with chaos}
Belonging to the same spherical layer the vector fields
$\mathbf{V}^{sing|1}$ defined in eq.s(\ref{sing1c1}-\ref{sing1c3})
and $\mathbf{V}^{sing|1,-2}$ can be linearly combined and we still
get a solution of Euler equations. Giving a much stronger
coefficient to the vertical field and a weaker one to chaotic one,
we obtain stream lines that are chaotic on the small scale but move
upward or downward on larger scales. A purely illustrative example
is provided by the Beltrami field:
\begin{equation}\label{marocello}
    \mathbf{V}^{mix}\, = \,\mathbf{V}^{vert|1,-2}\, +
    \,\frac{1}{50}\, \mathbf{V}^{sing|1}
\end{equation}
whose plot, sided by a snapshot of its streamlines is shown in fig.
\ref{escoselvaoscura}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=80mm]{mistofildo.png}
\includegraphics[width=80mm]{chaosverticale.png}
\end{center}
\caption{\it Plot of the Beltrami field in eq.\ref{marocello} sided
by a plot of 25 of its streamlines, originating from the fundamental
planar cell. As one realizes some are uprising randomized spirals,
some other descending such spirals. \label{escoselvaoscura}}
\end{figure}
What happens is best understood by looking at the analysis of a
single streamline shown in fig.\ref{vagabondo}.
\begin{figure}[!htb]
\begin{flushleft}
\includegraphics[width=50mm]{incertopede.png}
\includegraphics[width=50mm]{umisto.png}
\includegraphics[width=50mm]{vmisto.png}
\includegraphics[width=50mm]{rmisto.png}
\end{flushleft}
\caption{\it Plot of a single streamline starting at the point $\{0,
0, 0\}$ for the vector field defined in eq. (\ref{marocello}) and
displayed in  fig. \ref{escoselvaoscura}.
 \label{vagabondo}.Instead of climbing straight ahead the fluid elements winds a little bit around but finally reaches the top.}
\end{figure}
It is clear that the combination of the upward (or downward)
movement and the chaotic movement can be calibrated by the choice of
the relative parameters. Furthermore there are many more
chaotic-like fields in the same layer that can be utilized to
calibrate the relative behavior what we have shown being only an
example of the superposition of the two mechanisms.
\chapter{Conclusions}
In the previous chapters of the present report we have outlined and
presented the theoretical basis of the mechanism \textbf{Chaos from
symmetry}, emerging from the use of Beltrami fields as ingredients
of exact periodic solutions of the Navier Stokes equations. When the
compact space in which they occur is a three torus, as introduced in
eq. (\ref{metricT3}), these exact solutions are governed quite
efficiently by Group Theory. This is the fundamental message that is
not widely and fully appreciated neither among the differential
geometers and dynamical system theorists that give important
contributions to the field of mathematical hydrodynamics, nor among
the applied scientists doing numerical simulations and working in
CFD.
\par
Notwithstanding their long life Navier Stokes equations have few
exact solutions, the existing ones providing already a wide spectrum
of qualitatively different behaviors and enucleating the essential
point of difficulty that can be summarized as follows.
\par
As stressed in the introduction both conceptually and at the level
of applications, one would like to consider hydro flows that have at
least two scales, a \textit{macro scale} where we observe a
directional, reasonably ordered flow and a \textit{micro scale}
where the flow is instead chaotic. How to combine the two aspects
into exact solutions is the open unsolved problem.
\par
Vladimir Arnold unveiled since the years 70.s of the XXth century
the profound topological nature of chaotic behavior
\cite{arnoldus,ArnoldBook}. His theorem \ref{fundarnoltheo}
emphasizes the essential role of Beltrami vector fields that are
precisely what, after the work of one of us with A.Sorin of
2014-2015 \cite{Fre:2015mla}, can be precisely classified and
constructed in terms of Finite Group Theory. On the other hand
Beltrami fields have a natural relation, in the capacity of Reeb
fields, with the geometrical conception of \textit{contact
structures} on odd-dimensional manifolds. Contact Manifolds in
odd-dimensions  have symbiotic relations with symplectic manifolds
one dimension above and one dimension below and so does their
defining contact one-form; all that brings into the field of
hydrodynamics  the visions and the arguments of differential
geometry of symplectic related type. The group-theoretical
classification of Beltrami fields therefore reflects into a
group-theoretical classification of contact-structures and of their
allied even-manifolds. Contact structures are the deep root of
chaotic behavior being the geometric obstruction to the existence of
a foliation of the ambient manifold $\mathcal{M}$ in which the fluid
moves and foliations being, instead, the essential ingredient of
potential or laminar ordered flows.
\par
So the difficulty in reconciling the above mentioned two scale
regimes within one and the same exact solution of Navier Stokes
equation is not an occasional one rather it is a very much
conceptual antinomy.
\par
What are the possible strategic way out? Three have emerged that
might be combined together:
\begin{enumerate}
  \item Within the scope of the landscape approach to Beltrami fields one can
  superimpose motions that look like directional ones on larger
  scales,
  although they reveal themselves as winding spirals at smaller scales, with properly
  chaotic motions at short distances. From the point of view of
  contact structure Beltrami fields are necessary for chaos yet not
  viceversa. There are Beltrami fields that give rise to integral
  systems and cohexist on the same spherical layer with really
  chaos-generating Beltrami fields that typically are the most
  symmetric ones.
  \item Consider the new development of singular contact structures
  and singular Beltrami fields in so named $\mathfrak{b}$-manifolds.
  \item Reconsider the results of \cite{beltraspectra} where it was
  shown that solutions of Navier Sokes equations display the feature
  of weekly interacting Beltrami spectra.
\end{enumerate}
As we have shown in the present work the case 2) of the above list
that was initiated by the authors of
\cite{guglielminoconmiranda,unochiuseinteg,trasversozero,cardona2020universality,
cardona2021integrable,pollosingolare,cardone2019,miranda2021} has an
unsuspected strong relation with the group theoretical structure of
Beltrami fields that requires to be clarified in detail and is
potentially very powerful.
\par
Similarly the in depth analysis of the landscape properties in
search of algorithmic recipes for the optimal synthesis of  over all
directional flows with low scale chaotic flows requires appropriate
group theoretical investigations and also extensive surveys of the
landscape at large. For instance the classification of the 48
momentum classes of the cubic lattice achieved in \cite{Fre:2015mla}
has not yet been done for the hexagonal lattice. To this effect
implementation of the MATHEMATICA Codes on large powerful computers
would be quite appropriate.
\par
As for 3), the main observation of the authors of
\cite{beltraspectra} is that the Beltrami operator is a chiral one.
There are solutions with $\pm \lambda$ where $\lambda$ is an
admissible eigenvalue. Hence the conventional Fourier expansion of
any periodic field can be subdivided in two parts the Beltrami plus
and the Beltrami minus part (or anti-Beltrami). The authors observed
that Beltrami fields weakly interact amang themselves and one can
define and index of \textit{Beltramicity} per frequency that is
approximately conserved. This viewpoint can be very fertile in
dealing with our group theoretical construction of Beltrami fields.
In the landscape approach we are free to sum exact solutions only in
a definite spherical layer. If we mix different layers we obtain
objects that are not exact solutions. Yet the weak interaction
viewpoint suggests that such sums can approximate exact solutions in
a nice way and controllable way. This is obviously a very much
promising direction to be explored with great care in future
publications.
\par
 A fourth direction of development, still in its infancy,
which is worth perceiving concerns the reduction of the cubic
lattice with respect the tetrahedron subgroup $\mathrm{T_{12}}$ of
the octahedral group $\mathrm{O_{24}}$. Constructing Beltrami fields
in a fundamental tetrahedrical rather than cubic cell might lead to
a new quality of solutions and in particular one might study complex
three dimensional geometries in terms of symplexes.
\par
These are the principal directions of further research that are
suggested by the results of the present investigation. The
construction, testing and validation of the two MATHEMATICA CODES
described in the Appendices is to be regarded as an achievement
since they constitute the essential basis for all the further
investigations outlined above and the starting point for all
possible future applications.

\section*{Aknowledgements} The present scientific investigation has
been conducted within the framework of the Project \textit{ALMA
FLUIDA} partially financed by the Toscana Region as part of the
Consultancy Agreement between the Company ITALMATIC Presse e Stampi
s.r.l and the DISAT of Politecnico di Torino. The origin of this
investigation is to be traced back to a previous study of
Arnold-Beltrami Flows conducted in 2014-15 by one of the present
authors (P.F.), together with his long time collaborator and close
friend Alexander Sorin. P.F. would  like to express his gratitude to
A. Sorin for introducing him to this topic that now finds new life.
Last but not least we desire to express  our gratitude to our great
friend Sauro Additati for envisaging the whole scheme of the ALMA
FLUIDA project and to the Fredianis, father Adriano and son Roberto,
directing ITALMATIC Presse e Stampi, who have made this scientific
mission not only possible but also very pleasant due to their warm
and deep friendship. Last but not least it is a pleasure to express
our gratitude to Daniele Marchisio first of all for the amicable
relations that we have developed with him during these months since
the time when the ALMA FLUIDA project was firstly conceived and
constructed together, secondly for his excellent coordination of the
Politecnico consultancy still going on.
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redpajama_set_name: RedPajamaArXiv