text:
\section{Introduction}

The set of known maximally superintegrable systems on the
$N$-dimensional ($N$D) Euclidean space is very limited: it
comprises the isotropic harmonic oscillator with $N$ centrifugal
terms (the so-called Smorodinsky--Winternitz (SW) system
\cite{fris,evans2}), the Kepler--Coulomb (KC) problem with $(N-1)$
centrifugal barriers \cite{MAW} (and some symmetry-breaking
generalizations of it \cite{Pogo}), the
Calogero--Moser--Sutherland model \cite{CMS,SW,RM,Goner1} and some
systems with isochronous potentials~\cite{Gonera}. Both the SW and
the KC systems have  integrals quadratic in the momenta, and also
both of them have been generalized to spaces with non-zero
constant curvature
(see~\cite{VulpiLett,CRMVulpi,RS,PogosClass1,PogosClass2,KalninsH2,Higgs,Leemon,
Schrodingerdual,Schrodingerdualb,kiev}). In order to complete this
brief $N$D summary, Benenti systems on constant curvature spaces
have also to be considered \cite{Benenti}, as well as a maximally
superintegrable deformation of the SW system that was introduced
in \cite{Deform} by making use of quantum algebras.

More recently, the study of 2D and 3D superintegrable systems on
spaces with variable curvature has been
addressed~\cite{darboux1,darboux2,darboux3,darboux4,darboux5,darboux6,darboux7}.
The aim of this paper is to give a~general setting, based on
quantum deformations, for the explicit construction of certain
classes of superintegrable systems on $N$D spaces with
 variable curvature.

In order to f\/ix language conventions, we recall that an
 $N$D completely integrable Hamiltonian~$H^{(N)}$ is called {\it maximally superintegrable} (MS) if there exists
a set of $(2N-2)$ globally def\/ined  functionally independent
constants of the motion that Poisson-commute with~$H^{(N)}$. Among
them, at least two dif\/ferent subsets of $(N-1)$ constants in
involution can be found. In the same way, a system will be called
{\it quasi-maximally superintegrable} (QMS) if there are $(2N-3)$
integrals with the abovementioned properties. All MS systems are
QMS ones, and the latter have only one less integral than the
maximum possible number of functionally independent ones.

In this paper we present the construction of QMS systems on
variable curvature spaces which is just the quantum algebra
generalization of a recent approach to $N$D QMS systems on
constant curvature spaces that include the SW and KC as particular
cases \cite{BHletter}. Some of these variable curvature systems in
2D and 3D have been already studied (see \cite{plb,jpa2D,Checz}),
and we present here the most signif\/icant elements for their $N$D
generalizations. We will show that this scheme is quite
ef\/f\/icient in order to get explicitly a large family of QMS
systems. Among them, some specif\/ic choices for the Hamiltonian
can lead to a MS system, for which only the remaining integral has
to be explicitly found.

In the the next Section we will brief\/ly summarize the $N$D
constant curvature construction given in \cite{BHletter}, that
makes use of an $sl(2,\mathbb R)$ Poisson coalgebra symmetry. The
generic variable curvature approach will be obtained in Section 3
through a non-standard quantum deformation of an $sl(2,\mathbb R)$
Poisson coalgebra. Some explicit 2D and 3D spaces def\/ined
through free motion Hamiltonians will be given in Section 4, and
the $N$D generalization of them will be sketched in Section 5.
Section 6 is devoted to the introduction of some potentials that
generalize the KC and SW ones. A f\/inal Section including some
comments and open questions closes the paper.

\section[QMS Hamiltonians with  $sl(2,\mathbb R)$ coalgebra symmetry]{QMS Hamiltonians with  $\boldsymbol{sl(2,\mathbb R)}$ coalgebra symmetry}

Let us brief\/ly recall the main result of \cite{BHletter} that
provides an inf\/inite family of QMS Hamiltonians. We stress that,
although some of these Hamiltonians can be interpreted as motions
on spaces with constant curvature, this approach to QMS systems is
quite general, and also non-natural Hamiltonian systems (for
instance, those describing static electromagnetic f\/ields) can be
obtained.

\begin{theorem}[\cite{BHletter}]
Let $\{\>q,\>p \}=\{(q_1,\dots,q_N),(p_1,\dots,p_N)\}$ be $N$
pairs of canonical variables. The ND Hamiltonian
\begin{equation}
H^{(N)}={\cal H}\big(\>q^2,\tilde{\>p}^2,\>q\cdot\>p\big),
\label{hgen}
\end{equation}
with ${\cal H}$ any smooth function and
\begin{equation*}
 \>q^2=\sum_{i=1}^N q_i^2 , \qquad\    \tilde{\>p}^2=
    \sum_{i=1}^N \left(  p_i^2+\frac{b_i}{ q_i^2} \right)
\equiv \>p^2 +  \sum_{i=1}^N  \frac{b_i}{ q_i^2} ,  \qquad\
\>q\cdot\>p =
  \sum_{i=1}^N  q_i\, p_i ,
\end{equation*}
where $b_i$ are arbitrary real parameters, is QMS. The   $(2N-3)$
functionally independent and ``universal" integrals of  motion are
explicitly given by
\begin{gather}
 C^{(m)}= \sum_{1\leq i<j}^m \left\{ ({q_i}{p_j} -
{q_j}{p_i})^2 + \left(
b_i\frac{q_j^2}{q_i^2}+b_j\frac{q_i^2}{q_j^2}\right)\right\}
+\sum_{i=1}^m b_i , \nonumber\\
C_{(m)}= \sum_{N-m+1\leq i<j}^N \left\{ ({q_i}{p_j} -
{q_j}{p_i})^2 + \left(
b_i\frac{q_j^2}{q_i^2}+b_j\frac{q_i^2}{q_j^2}\right)\right\}
+\sum_{i=N-m+1}^N b_i  , \label{cinfm}
\end{gather}
where $m=2,\dots, N$ and $C^{(N)}=C_{(N)}$.  Moreover, the sets of
$N$ functions $\{H^{(N)},C^{(m)}\}$ and $\{H^{(N)},C_{(m)}\}$
$(m=2,\dots, N)$ are in involution.
\end{theorem}

The proof of this general result is based on the observation that,
for any choice of the function ${\cal H}$, the Hamiltonian
$H^{(N)}$ has an $sl(2,\mathbb R)$ Poisson coalgebra
symmetry~\cite{BR} generated by the following Lie--Poisson
brackets and comultiplication map:
\begin{gather}
 \{J_3,J_+\}=2 J_+     ,\qquad
\{J_3,J_-\}=-2 J_- ,\qquad \{J_-,J_+\}=4 J_3  , \label{ba}
\\
\Delta(J_l)=  J_l \otimes 1+ 1\otimes J_l ,\qquad l=+,-,3.
\label{bb}
\end{gather}
The Casimir function for $sl(2,\mathbb R)$ reads
\begin{equation}
{\cal C}=  J_- J_+ -J_3^2  . \label{bc}
\end{equation}

 In fact, the coalgebra approach~\cite{BR} provides an
$N$-particle symplectic realization of $sl(2,\mathbb R)$  through
the $N$-sites coproduct of (\ref{bb}) living on $sl(2,\mathbb
R)\otimes  \cdots^{N)}\otimes sl(2,\mathbb R)$~\cite{Deform}:
\begin{equation}
 J_-=\sum_{i=1}^N q_i^2\equiv \>q^2 ,\qquad    J_+=
    \sum_{i=1}^N \left(  p_i^2+\frac{b_i}{ q_i^2} \right)
\equiv \>p^2 +  \sum_{i=1}^N  \frac{b_i}{ q_i^2}  ,\qquad J_3=
  \sum_{i=1}^N  q_i p_i\equiv \>q\cdot\>p  ,
\label{be}
\end{equation}
where $b_i$ are $N$ arbitrary real parameters. This means that
the $N$-particle generators (\ref{be}) fulf\/il the  commutation
rules (\ref{ba}) with respect to the canonical Poisson bracket. As
a consequence of the coalgebra approach, these generators Poisson
commute with the $(2N-3)$ functions (\ref{cinfm}) given by the
sets $C^{(m)}$ and $C_{(m)}$, which are obtained, in this order,
from the ``left'' and ``right'' $m$-th coproducts of the Casimir
(\ref{bc}) with $m=2,3,\dots,N$ (see \cite{CRMAngel} for details).
Therefore, any   arbitrary function ${\cal H}$ def\/ined on the
$N$-particle symplectic realization of $sl(2,\mathbb R)$
(\ref{be}) is of the form (\ref{hgen}), that is,
\[
{H}^{(N)}= {\cal H}\left(J_-,J_+,J_3\right) ={\cal
H}\left(\>q^2,\>p^2+\sum_{i=1}^N\frac{b_i}{q_i^2},\>q\cdot\>p
\right),
\]
 and def\/ines a QMS
Hamiltonian system that Poisson-commutes with all the ``universal
integ\-rals''~$C^{(m)}$ and $C_{(m)}$.

Notice that for arbitrary $N$ there is a single constant of the
motion left  to assure   maximal superintegrability. In this
respect, we   stress that some specif\/ic choices of ${\cal H}$
comprise maximally superintegrable systems as well, but the
remaining integral does not come from the coalgebra symmetry and
has to be deduced by making use of alternative procedures.

Let us now give some explicit examples of this construction.

\subsection{Free motion on Riemannian spaces of constant curvature}

It is immediate to realize that the  kinetic energy  ${\cal T}$ of
a particle on the $N$D  Euclidean space ${{\bf E}^N}$ directly
arises through the generator $J_+$ in the symplectic realization
(\ref{be})  with all $b_i=0$:
\begin{equation*}
 {\cal H} = {\cal T}=\frac 12 J_+ =\frac 12\, {\>p}^2 .

\end{equation*}

Now the interesting point is that the kinetic energy
  on $N$D Riemannian spaces with constant curvature $\kappa$ can be expressed
in Hamiltonian form as a function of the $N$D symplectic
realization of the $sl(2,\mathbb R)$ generators  (\ref{be}).  In
fact, this can be done in two dif\/ferent ways \cite{BHletter}:
\begin{gather}
\displaystyle{ {\cal H}^{\rm P}={\cal T}^{\rm P}=\frac{1}{2}\left(
1+\kappa J_-\right)^2 J_+=
\frac{1}{2}\left( 1+\kappa \>q^2\right)^2 \>p^2} ,\nonumber\\
\displaystyle{ {\cal H}^{\rm B}={\cal T}^{\rm B}=\frac{1}{2}\left(
1+\kappa J_-\right)\left(  J_+ +\kappa J_3^2\right)= \frac{1}{2}(1+\kappa
\>q^2)\left( \>p^2+\kappa (\>q\cdot \>p)^2 \right) }. \label{dd}
\end{gather}

The function ${\cal H}^{\rm P}$ is just the kinetic energy for a
free particle  on  the spherical ${\bf S}^N$ ($\kappa>0$) and
hyperbolic ${\bf H}^N$    ($\kappa<0$) spaces  when this is expressed
in terms of Poincar\'e coordinates $\>q$ and canonical momenta
$\>p$ (coming from a stereographic projection in ${\mathbb
R^{N+1}}$);  on the other hand ${\cal H}^{\rm B}$ corresponds to
Beltrami coordinates and momenta (central projection). By
construction, both Hamiltonians are QMS ones since they
Poisson-commute with the integrals (\ref{cinfm}).

\subsection{Superintegrable potentials on Riemannian spaces of constant curvature}

QMS potentials ${\cal V}$ on constant curvature spaces can now be
constructed by adding some suitable functions depending on $J_-$
to (\ref{dd}) and by considering arbitrary centrifugal  terms that
come from symplectic realizations of the $J_+$ generator with
generic $b_i$'s:
\begin{equation*}
{\cal H}={\cal T}(J_+,J_-,J_3) + {\cal V}(J_-).
\end{equation*}
The Hamiltonians that we will obtain in this way are the curved
counterpart of the Euclidean systems, and through dif\/ferent
values of the curvature $\kappa$ we will simultaneously cover the
cases~${\bf S}^N$ ($\kappa>0$), ${\bf H}^N$    ($\kappa<0$), and ${\bf
E}^N$
 ($\kappa=0$).

In order to motivate the choice of the potential functions ${\cal
V}(J_-)$, it is important to recall that in the constant curvature
analogues of the oscillator and KC problems the Euclidean radial
distance $r$ is just replaced by the function $
\frac{1}{\sqrt{\kappa}}\tan(\sqrt{\kappa}\,r) $ (see \cite{BHletter} for
the expression of this quantity in terms of Poincar\'e and
Beltrami coordinates). Also, for the sake of simplicity, the
centrifugal terms coming from the symplectic realization with
arbitrary $b_i$ will be expressed in ambient coordinates
$x_i$~\cite{BHletter}:
\[
{\mbox{Poincar\'e:}}\quad  x_i=  \frac{2 q_i}{1+ \kappa\>q^2};\qquad
 {\mbox{Beltrami:}} \quad
x_i=\frac{q_i}{\sqrt{1+ \kappa \>q^2}}.
\]

Special  choices for ${\cal V}(J_-)$ lead to the following
systems, that are always expressed in both Poincar\'e and Beltrami
phase spaces:
\medskip

\noindent $\bullet$ {\em A curved Evans system}. The constant
curvature generalization of a 3D Euclidean system with radial
symmetry~\cite{Evansa} would be given by
\begin{gather}
\displaystyle{ {\cal H}^{\rm P}={\cal T}^{\rm P}+{\cal V} \left(
\frac{4 J_-}{ (1-\kappa J_-)^2 } \right) = \frac12 {\left( 1+\kappa
\>q^2\right)^2 \>p^2}}+{\cal V}\left( \frac{4\>q^2}{(1-\kappa\>q^2)^2}
\right) + \sum_{i=1}^N\frac{2b_i}{x_i^2},
\nonumber\\
\displaystyle{ {\cal H}^{\rm B}={\cal T}^{\rm B}+ {\cal
V}\left(J_-\right)= \frac{1}{2}(1+\kappa \>q^2)\left( \>p^2+\kappa
(\>q\cdot \>p)^2 \right) +{\cal V}\left(
\>q^2\right)+\sum_{i=1}^N\frac{b_i}{2 x_i^2}}, \label{eb}
\end{gather}
where ${\cal V}$ is an arbitrary smooth function that determines
the  central potential;   the specif\/ic  dependence  on $J_-$ of
${\cal V}$ corresponds to the square of the radial distance in
each coordinate system.

\medskip
\noindent $\bullet$ {\em The curved Smorodinsky--Winternitz
 system}~\cite{VulpiLett,CRMVulpi,RS,PogosClass1,PogosClass2,KalninsH2}.
Such a system is just the Higgs oscillator~\cite{Higgs,Leemon}
with angular frequency $\omega$ (that arises as the argument of
${\cal V}$ in (\ref{eb})) plus the corresponding centrifugal
terms:
\begin{gather*}
 {\cal H}^{\rm P}={\cal T}^{\rm P}+   \frac{4 \omega^2
J_-}{ (1-\kappa J_-)^2 } = \frac12 {\left( 1+\kappa \>q^2\right)^2 \>p^2}+
\frac{4\omega^2 \>q^2}{(1-\kappa\>q^2)^2}   +
\sum_{i=1}^N\frac{2b_i}{x_i^2} ,
\nonumber\\
{\cal H}^{\rm B}={\cal T}^{\rm B}+\omega^2 J_- = \frac{1}{2}(1+\kappa
\>q^2)\left( \>p^2+\kappa (\>q\cdot \>p)^2 \right) + \omega^2
\>q^2+\sum_{i=1}^N\frac{b_i}{2 x_i^2}.
\end{gather*}
This is a     MS Hamiltonian and the remaining constant of the
motion can be chosen from any of  the following $N$ functions:
\begin{gather*}
{\cal I}_i^{\rm P}=\left( p_i(1-\kappa \>q^2) + 2\kappa (\>q\cdot \>p) q_i
\right)^2+ \frac{8  \omega^2 q_i^2}{(1-\kappa\>q^2)^2}   +
{b_i}\,
\frac{(1-\kappa\>q^2)^2} { q_i^2}  ,\nonumber\\
 {\cal I}_i^{\rm B}=\left( p_i+\kappa (\>q\cdot \>p) q_i \right)^2+2
\omega^2  q_i^2+  {b_i}/{q_i^2},\qquad i=1,\dots,N.
\end{gather*}

\noindent $\bullet$ {\em A  curved generalized Kepler--Coulomb
system}~\cite{RS, PogosClass1,PogosClass2,Schrodingerdual,
Schrodingerdualb,kiev}. The     curved KC potential with real
constant $k$ together with $N$ centrifugal terms would be given by
\begin{gather*}  {\cal H}^{\rm P}={\cal T}^{\rm P}-k \left(\frac{4
J_-}{ (1-\kappa J_-)^2 } \right)^{-1/2}= \frac12 {\left( 1+\kappa
\>q^2\right)^2 \>p^2}- k \, \frac{(1-\kappa\>q^2)}   {2
\sqrt{\>q^2}}+ \sum_{i=1}^N\frac{2b_i}{x_i^2},
\nonumber\\
{\cal H}^{\rm B}={\cal T}^{\rm B} -k  J_-^{-1/2}= \frac{1}{2}(1+\kappa
\>q^2)\left( \>p^2+\kappa (\>q\cdot \>p)^2 \right) -\frac{k}{
\sqrt{\>q^2 }}+\sum_{i=1}^N\frac{b_i}{2 x_i^2} .
\end{gather*}
This is again  a MS system provided that, at least, one
$b_i=0$. In this case the remaining   constant of the motion
turns out to be
\begin{gather}
 {\cal L}_i^{\rm P}=\sum_{l=1 }^N \left(
p_l(1-\kappa \>q^2) + 2\kappa (\>q\cdot \>p) q_l \right) (q_l p_i-q_i p_l)
+ \frac{k  q_i}{2\sqrt{\>q^2}}-
\sum_{l=1;l\ne i}^N b_l\,\frac{ q_i(1-\kappa \>q^2)}{ q_l^2}   ,\nonumber\\
 {\cal L}_i^{\rm B}=\sum_{l=1 }^N  \left(
p_l+\kappa (\>q\cdot \>p) q_l \right)   (q_l p_i-q_i p_l)
 +\frac{ k q_i}{\sqrt{\>q^2}}-
\sum_{l=1;l\ne i}^N b_l\,\frac{ q_i}{ q_l^2}  . \label{ceex}
\end{gather}
If another $b_j=0$, then ${\cal L}_j^{\rm P,B}$ is also a new
constant of the motion. In this way the proper curved KC
system~\cite{Schrodinger} (with   all the $b_i$'s equal to
zero) is obtained, and in that case (\ref{ceex}) are just
the~$N$~components of the  Laplace--Runge--Lenz vector on ${\bf
S}^N$ $(\kappa>0)$ and ${\bf H}^N$ $(\kappa<0)$.

We also stress that all these examples share the {\em same} set of
constants of the motion  (\ref{cinfm}), although the geometric
meaning of the canonical coordinates and momenta can be
dif\/ferent.

\section[QMS Hamiltonians with quantum deformed $sl(2,\mathbb R)$ coalgebra symmetry]{QMS Hamiltonians with quantum deformed $\boldsymbol{sl(2,\mathbb R)}$ coalgebra\\ symmetry}

Here  we will show   that a generalization of the construction
presented in the previous Section  can be obtained through a
quantum deformation of   $sl(2,\mathbb R)$,  yielding  QMS systems
for
 certain spaces with variable curvature. Let us now state the general
statement that provides a superintegrable deformation of Theorem
1.

\begin{theorem}
Let $\{\>q,\>p \}=\{(q_1,\dots,q_N),(p_1,\dots,p_N)\}$ be $N$
pairs of canonical variables. The N{\rm{D}} Hamiltonian
\begin{equation}
H^{(N)}_z={\cal
H}_z\left(\>q^2,\tilde{\>p}_z^2,(\>q\cdot\>p)_z\right),
\label{hgenb}
\end{equation}
where ${\cal H}_z$ is any smooth function and
\begin{gather*}
\>q^2= \sum_{i=1}^N q_i^2,\qquad
 \tilde{\>p}_z^2=\sum_{i=1}^N
\left( \frac {\sinh z q_i^2}{z q_i^2} \, p_i^2  +\frac{z
b_i}{\sinh z
q_i^2} \right) {\rm e}^{z K_i^{(N)}(q^2) },\\%\label{sympz}\\
(\>q\cdot\>p)_z=\sum_{i=1}^N \frac {\sinh z q_i^2}{z q_i^2} \,
q_ip_i \, {\rm e}^{z K_i^{(N)}(q^2) }  , \nonumber
\end{gather*}
with
\begin{equation}
  K_i^{(h)}(q^2)=  - \sum_{k=1}^{i-1}  q^2_k+
\sum_{l=i+1}^h   q^2_l , \label{kki}
\end{equation}
is QMS for any choice of the function ${\cal H}$ and for arbitrary
real parameters $b_i$. The $(2N-3)$ functionally independent and
``universal" integrals of the motion
   are   given by
\begin{gather}
C_z^{(m)}= \sum_{1\leq i<j}^m{Q_{ij}^z}\,{\rm e}^{ z
K_{ij}^{(m)}(q^2)} + \sum_{i=1}^m b_i \, {\rm e}^{2 z
K_{i}^{(m)}(q^2)},\nonumber\\
C_{z,(m)}= \sum_{N-m+1\leq i<j}^N{Q_{ij}^z}\,{\rm e}^{ z
\tilde{K}_{ij}^{(N-m+1)}(q^2)} + \sum_{i=N-m+1}^N b_i\,  {\rm e}^{2
z \tilde{K}_{i}^{(N-m+1)}(q^2)}, \label{fk}
\end{gather}
where  $m=2,\dots, N$,  $C_z^{(N)}=C_{z,(N)}$, and
\begin{gather*}
 K_{ij}^{(h)}(q^2)  = K_i^{(h)}(q^2)  +  K_j^{(h)}(q^2)
  =    - 2\sum_{k=1}^{i-1}   q^2_k  -    q^2_i  +    q^2_j  +
2\sum_{l=j+1}^h   q^2_l  ,\nonumber\\
\tilde{K}_i^{(h)}(q^2)=  - \sum_{k=h}^{i-1}  q^2_k+
\sum_{l=i+1}^N   q^2_l,\nonumber\\
 \tilde{K}_{ij}^{(h)}(q^2)  = \tilde{K}_i^{(h)}(q^2)  +  \tilde{K}_j^{(h)}(q^2)
  =    - 2\sum_{k=h}^{i-1}   q^2_k  -    q^2_i  +    q^2_j  +
2\sum_{l=j+1}^N   q^2_l, \nonumber\\
Q_{ij}^z=\left\{\frac {\sinh z q_i^2}{z  q_i^2}\, \frac {\sinh z
q_j^2}{z q_j^2} \left({q_i}{p_j} - {q_j}{p_i}\right)^2   +\left(
b_i\, \frac {\sinh z q_j^2}{\sinh z q_i^2} + b_j\, \frac
{\sinh z q_i^2}{\sinh z q_j^2} \right)\right\},
\end{gather*}
with $i<j$.  Moreover, the sets of $N$ functions
$\{H^{(N)}_z,C^{(m)}_z\}$ and $\{H^{(N)}_z,C_{z,(m)}\}$
$(m=2,\dots, N)$ are in involution.
\end{theorem}

\subsection{The proof}

The proof  is based on the fact that, for any choice of the
function ${\cal H}$, the Hamiltonian $H^{(N)}_z$ has a~deformed
Poisson coalgebra symmetry, $sl_z(2,\mathbb R)$, coming (under a
certain   symplectic realization)  from the  non-standard quantum
deformation of    $sl(2,\mathbb R)$~\cite{Ohn,CP} where $z$ is the
deformation parameter $(q={\rm e}^z)$. If we perform  the   limit
$z\to 0$ in all the results given in Theorem 2, we shall exactly
recover Theorem 1.
  Here we
sketch the main steps of this construction, referring   to
\cite{Deform,CRMAngel} for further details.

We recall that the non-standard $sl_z(2,\mathbb R)$ Poisson
coalgebra is given by the following deformed Poisson brackets and
coproduct~\cite{Deform}:
\begin{gather}
\{J_3,J_+\}=2 J_+ \cosh zJ_-  ,\qquad
 \{J_3,J_-\}=-2\,\frac {\sinh zJ_-}{z} ,\qquad
 \{J_-,J_+\}=4 J_3 ,
\label{baa}
\\
  \Delta_z(J_-)=  J_- \otimes 1+
1 ,\qquad  \Delta_z(J_l)=J_l \otimes {\rm e}^{z J_-} + {\rm e}^{-z
J_-} \otimes J_l ,\qquad l=+,3. \label{ggcc}
\end{gather}
The Casimir function for $sl_z(2,\mathbb R)$ reads
\begin{equation}
{\cal C}_z=  \frac {\sinh zJ_-}{z}\, J_+ -J_3^2 . \label{gc}
\end{equation}

A  one-particle symplectic realization of (\ref{baa}) is given by
\begin{equation*}
J_-^{(1)}=q_1^2  ,\qquad
 J_+^{(1)}=\frac {\sinh z q_1^2}{z q_1^2}\, p_1^2 +
\frac{z b_1}{\sinh z q_1^2},\qquad J_3^{(1)}=\frac {\sinh z
q_1^2}{z q_1^2}\, q_1 p_1 ,
\end{equation*}
where $b_1$ is a real parameter that labels the representation
through ${\cal C}_z=b_1$.

Now the essential point is the fact that the coalgebra
approach~\cite{BR} provides the corresponding $N$-particle
symplectic realization of $sl_z(2,\mathbb R)$  through the
$N$-sites coproduct of (\ref{ggcc}) living on $sl_z(2,\mathbb
R)\otimes  \cdots^{N)}\otimes sl_z(2,\mathbb R)$~\cite{Deform}:
\begin{gather}
J_-^{(N)}= \sum_{i=1}^N q_i^2 \equiv \>q^2 ,\qquad
J_3^{(N)}=\sum_{i=1}^N \frac {\sinh z q_i^2}{z q_i^2} \, q_ip_i \,
{\rm e}^{z K_i^{(N)}(q^2) }
\equiv (\>q\cdot\>p)_z ,\nonumber \\
 J_+^{(N)}=\sum_{i=1}^N
\left( \frac {\sinh z q_i^2}{z q_i^2} \, p_i^2  +\frac{z
b_i}{\sinh z q_i^2} \right)  {\rm e}^{z K_i^{(N)}(q^2)
}\equiv \tilde{\>p}_z^2 , \label{zsymp}
\end{gather}
where $K_i^{(N)}(q^2)$ is def\/ined in (\ref{kki}) and $b_i$
are $N$ arbitrary real parameters that label the representation on
each ``lattice'' site. This means that the $N$-particle generators
(\ref{zsymp}) fulf\/il the  commutation rules (\ref{baa}) with
respect to the   canonical Poisson bracket
\[
\{f,g\}=\sum_{i=1}^N\left(\frac{\partial f}{\partial q_i}
\frac{\partial g}{\partial p_i} -\frac{\partial g}{\partial q_i}
\frac{\partial f}{\partial p_i}\right).
\]
Therefore the Hamiltonian  (\ref{hgenb}) is obtained through an
arbitrary smooth function
  ${\cal H}_z$ def\/ined on the
$N$-particle symplectic realization of the generators of
$sl_z(2,\mathbb R)$:
\begin{equation}
{H}^{(N)}_z= {\cal H}_z\big(J_-^{(N)},J_+^{(N)},J_3^{(N)}\big)
={\cal H}_z\left(\>q^2,\tilde{\>p}_z^2,(\>q\cdot\>p)_z \right).
\label{hamham}
\end{equation}

By construction~\cite{BR}, the functions (\ref{zsymp}) Poisson
commute with the $(2N-3)$ functions~(\ref{fk}) given by the sets
$C^{(m)}_z$ and $C_{z,(m)}$, which are obtained from the ``left"
and ``right" $m$-th copro\-ducts of the Casimir (\ref{gc}) with
$m=2,3,\dots,N$~\cite{CRMAngel}. For instance, the $C_z^{(m)}$
integrals are nothing but
\begin{equation*}
C_z^{(m)}= \frac {\sinh zJ_-^{(m)}}{z}\, J_+^{(m)} -
\big(J_3^{(m)}\big)^2  ,
\end{equation*}
and the right ones $C_{z,(m)}$ can be obtained through an
appropriate permutation of the labelling of the lattice sites
(note that these integrals depend on the canonical coordinates
running from $(N-m+1)$ up to $N$). Thus ${H}^{(N)}_z$  Poisson
commutes with the $(2N-3)$ integrals and, furthermore, the
coalgebra symmetry also ensures that each of the subsets
$\{C_z^{(2)}, \dots ,C_z^{(N)},{H}^{(N)}_z \}$ and
$\{C_{z,(2)},\dots ,C_{z,(N)},{H}^{(N)}_z \}$ consists of  $N$
functions in involution.

In order to prove the functional independence of the   $2N-2$
functions $\{ C_z^{(2)},C_z^{(3)},\dots ,C_z^{(N)}\equiv
C_{z,(N)}, C_{z,(N-1)},\dots ,C_{z,(2)},{H}^{(N)}_z \}$ it
suf\/f\/ices to realize that such functions are just deformations
in the deformation parameter $z$ of the $sl(2,\mathbb R)$
integrals given by (\ref{cinfm}), and the latter (which are
recovered when $z\to 0$) are indeed functionally independent.

Thus, we  conclude that any   arbitrary function ${\cal H}_z$
(\ref{hamham}) def\/ines a QMS Hamiltonian system.

\subsection[The $N=2$ case]{The $\boldsymbol{N=2}$ case}

In order to illustrate the previous construction, let us
explicitly write the $2$-particle symplectic realization of
$sl_z(2,\mathbb R)$ (\ref{zsymp}):
\begin{gather*}
 J_-^{(2)}=q_1^2+q_2^2  , \qquad J_3^{(2)}=
 \frac {\sinh z q_1^2}{z q_1^2 } \, {\rm e}^{z q_2^2}   q_1 p_1  +
 \frac {\sinh z q_2^2}{z q_2^2 }\, {\rm e}^{-z q_1^2}    q_2 p_2  ,\nonumber\\
 J_+^{(2)}=
  \frac {\sinh z q_1^2}{z q_1^2}\,  {\rm e}^{z q_2^2}   p_1^2   +
 \frac {\sinh z q_2^2}{z q_2^2} \,  {\rm e}^{-z q_1^2}   p_2^2 +\frac{z
b_1}{\sinh z q_1^2}\,{\rm e}^{z q_2^2} + \frac{z
b_2}{\sinh z q_2^2} \,  {\rm e}^{-z q_1^2} .
\end{gather*}
In this case there is a single (left and right) constant of the
motion:
\begin{equation*}
C_z^{(2)}= \frac {\sinh zJ_-^{(2)}}{z}\, J_+^{(2)} -
\big(J_3^{(2)}\big)^2  .
\end{equation*}
After some straightforward computations this integral can be
expressed as
\begin{gather}
   C^{(2)}_z=
\frac {\sinh z q_1^2}{z  q_1^2}\, \frac {\sinh z q_2^2}{z q_2^2}
\left({q_1}{p_2} - {q_2}{p_1}\right)^2  {\rm e}^{ z (q_2^2 -
q_1^2)} +  b_1  {\rm e}^{2 z q_2^2 }+  b_2  {\rm e}^{-2 z
q_1^2 }\cr \phantom{C^{(2)}_z=}{}+ \left( b_1 \frac {\sinh z
q_2^2}{\sinh z q_1^2} + b_2 \frac {\sinh z q_1^2}{\sinh z
q_2^2} \right)  {\rm e}^{ z (q_2^2 - q_1^2)} . \label{casdos}
\end{gather}
By construction, this constant of the motion will Poisson-commute
with all the Hamiltonians
\[
{H}^{(2)}_z= {\cal H}_z\left(J_-^{(2)},J_+^{(2)},J_3^{(2)}\right).
\]

Note that in the $N=2$ case quasi-maximal superintegrability means
only integrability, i.e., the only constant given by   Theorem 2
is just $C_z^{(2)}\equiv C_{z,(2)}$; this fact does not exclude
that there could be some specif\/ic choices for ${\cal H}_z$ for
which an additional integral does exist. When $N\ge 3$, Theorem 2
will always provide QMS Hamiltonians.

\section{Free motion on 2D and 3D curved manifolds}

\subsection{2D curved manifolds}

Throughout this Section we will consider {only free motion}.
Therefore we shall take the symplectic realization with
$b_1=b_2=0$ in order to avoid centrifugal potential terms.
In general, we can consider an inf\/inite family of {\em
integrable} (and quadratic
 in the momenta) free $N=2$ motions with $sl_z(2,\mathbb R)$ coalgebra symmetry
through Hamiltonians of the type
\begin{equation}
 {H}^{(2)}_z =\frac 12 J_+^{(2)} f\bigl(zJ_-^{(2)}\bigr),
 \label{bbfff}
\end{equation}
where $f$ is an arbitrary smooth function such that
 $\displaystyle{\lim_{z\to 0}f\bigl(zJ_-^{(2)}\bigr)=1}$, that is,
$\lim\limits_{z\to 0}{H}^{(2)}_z=\frac 12 (p_1^2+p_2^2)$.  We
shall explore in the sequel some specif\/ic choices for $f$, and
we shall analyse the spaces generated by them.

\subsubsection{An integrable case}

Of course, the
 simplest choice will be just to set $f\equiv 1$ \cite{plb}:
\begin{equation}
{\cal H}^{\rm \rm I}_z=\frac 12
 J_+^{(2)}=\frac12 \left( \frac {\sinh z
 q_1^2}{z q_1^2} \, {\rm e}^{z q_2^2}  p_1^2   +
 \frac {\sinh z q_2^2}{z q_2^2} \, {\rm e}^{-z q_1^2}  p_2^2  \right)   .
 \label{bg}
\end{equation}
Hence the {kinetic energy} ${\cal T}^{\rm \rm I}_z(q_i,p_i)$ coming
from
 ${\cal H}^{\rm \rm I}_z$  is
\begin{equation}
 {\cal T}^{\rm \rm I}_z(q_i,\dot q_i)=\frac 12 \left(\frac
 {z q_1^2}{\sinh z q_1^2} \, {\rm e}^{-z q_2^2} \dot q_1^2   +
 \frac {z q_2^2}{\sinh z q_2^2} \, {\rm e}^{z q_1^2} \dot q_2^2
 \right),
 \label{ca}
\end{equation}
 and def\/ines a geodesic f\/low on a 2D
 Riemannian space with signature
 diag$(+,+)$ and metric given by:
\begin{equation}
 {\rm d} s_I^2=\frac {2z q_1^2}{\sinh z
 q_1^2} \, {\rm e}^{-z q_2^2} \,{\rm d} q_1^2   +
  \frac {2 z q_2^2}{\sinh z q_2^2} \, {\rm e}^{z q_1^2}\, {\rm d} q_2^2  .
 \label{cc}
\end{equation}
 The Gaussian curvature $K$ for this space can be computed through
\begin{equation*}
 K=\frac{-1}{\sqrt{g_{11} g_{22}}}\left\{ \frac{\partial}{\partial
 q_1} \left( \frac{1}{\sqrt{g_{11}}} \frac{\partial
 \sqrt{g_{22}}}{\partial q_1}
 \right)+
 \frac{\partial}{\partial
 q_2} \left( \frac{1}{\sqrt{g_{22}}} \frac{\partial
 \sqrt{g_{11}}}{\partial q_2}
 \right)\right\} ,
\end{equation*}
and turns out to be {non-constant and {\em negative}}:
\begin{equation*}
 K(q_1,q_2;z)=-   z \sinh\left(z(q_1^2+q_2^2) \right).

\end{equation*}
Therefore, the
 underlying 2D space is of {hyperbolic  type} and
 endowed with  a {``radial" symmetry}.

Let us now consider the following change of coordinates that
includes a new parameter $\lambda_2\ne 0$:
\begin{gather*}
 \cosh(\lambda_1 \rho)=\exp\left\{z(q_1^2+q_2^2)\right\},\qquad
  \sin^2(\lambda_2 \theta)=\frac{\exp\left\{2z q_1^2
 \right\}-1}{\exp\left\{2z(q_1^2+q_2^2)\right\}-1},
\end{gather*}
 where $z=\lambda_1^2$  and $\lambda_2$ can take either a real or a
 pure imaginary value.
Note that the new variable $ \cosh (\lambda_1 \rho)$ is a
collective variable, a function of $\Delta(J_-)$; its role will be
specif\/ied later. On the other hand, the zero-deformation limit
$z\to 0$  is in fact the f\/lat limit $K \to 0$, since in this
limit
\begin{gather*}
\rho \to 2(q_1^2+q_2^2), \qquad \sin ^2 (\lambda_2 \theta) \to
\frac{q_1^2}{q_1^2 + q_2^2}.
\end{gather*}
Thus $\rho$ can be interpreted as  a {radial coordinate} and $\theta$
 is  {either a circular ($\lambda_2$ real) or a hyperbolic angle}  ($\lambda_2$ imaginary). Notice
that in the latter case, say $\lambda_2={\rm i}$, the coordinate $q_1$
is imaginary and can be written as $q_1={\rm i} \tilde q_1$ where
$\tilde q_1$ is a real coordinate; then $\rho \to 2(q_2^2-\tilde
q_1^2)$ which corresponds to a relativistic radial distance.
Therefore the introduction of the additional parameter $\lambda_2$
will allow us to obtain Lorentzian metrics.

 In
this new coordinates, the metric (\ref{cc}) reads
\begin{equation*}
 {\rm d} s_I^2=\frac {1}{\cosh(\lambda_1 \rho)}
 \left( {\rm d} \rho^2  +\lambda_2^2\,\frac{\sinh^2(\lambda_1 \rho)}{\lambda_1^2} \, {\rm d}
 \theta^2  \right) =\frac {1}{\cosh(\lambda_1 \rho)}\,{\rm d} s_0^2,
 \label{cg}
\end{equation*}
where ${\rm d} s_0^2$ is just the metric of the 2D Cayley--Klein spaces
 in terms of geodesic polar coordinates~\cite{ramon,Conf}
  provided that we identify $z=\lambda_1^2\equiv -\kappa_1$ and
 $\lambda_2^2\equiv \kappa_2$; hence $\lambda_2$  determines
the signature of the metric. The {Gaussian curvature} turns out to
be
\begin{equation*}
 K(\rho)=-\frac 12 \lambda_1^2 \,\frac{\sinh^2(\lambda_1 \rho)}{\cosh(\lambda_1
 \rho)} .

\end{equation*}
In this way we  f\/ind the {following   spaces}, whose main
properties are summarized in Table 1:

\begin{itemize}\itemsep=0pt

 \item[$\bullet$]  When  $\lambda_2$ is real, we get a  2D {deformed
 sphere} ${\bf S}^2_z$ $(z<0)$,    and  a {deformed
 hyperbolic or  Lobachewski space} ${\bf H}^2_z$ $(z>0)$.

 \item[$\bullet$]  When  $\lambda_2$ is imaginary, we obtain a {deformation
 of   the (1+1)D   anti-de Sitter spacetime} ${\bf AdS}_z^{1+1}$ $(z<0)$ and
of the   {de Sitter one} ${\bf dS}_z^{1+1}$
 $(z>0)$.

 \item[$\bullet$] In the non-deformed case $z\to 0$,  the Euclidean
 space
 ${\bf E}^2$
 ($\lambda_2$ real) and  Minkowskian spacetime
 ${\bf M}^{1+1}$ ($\lambda_2$ imaginary) are recovered.

 \end{itemize}

 \begin{table}
 {\footnotesize
  \noindent
 \caption{{{Metric and Gaussian curvature of the 2D spaces with $sl_z(2,\mathbb R)$
coalgebra symmetry for
 dif\/ferent values of the deformation parameter $z=\lambda_1^2$ and
 signature parameter $\lambda_2$.}}}
 \label{table1}
 \medskip
 \noindent\hfill
 $$
\!\!\!\!\!\!\!\!\!\!\back  \begin{array}{ll}
 \hline
 \\[-6pt]
 {\mbox {2D deformed Riemannian spaces}}&\quad{\mbox  {$(1+1)$D
 deformed relativistic spacetimes}}\\
 [4pt]
 \hline
 \\
 [-6pt]
 \mbox {$\bullet$ {Deformed sphere} ${\bf S}^2_z$}&\quad\mbox
{$\bullet$
 {Deformed anti-de Sitter spacetime} ${\bf AdS}^{1+1}_z$}\\[4pt] z=-1;\
 (\lambda_1,\lambda_2)=({\rm i},1)&\quad z=-1;\ (\lambda_1,\lambda_2)=({\rm i},{\rm
 i})\\[4pt]
 \displaystyle{{\rm d} s^2 =\frac{1}{\cos \rho}\left( {\rm d} \rho^2+\sin^2
 \rho\,{\rm d}\theta^2
 \right)} &\quad
 \displaystyle{{\rm d} s^2 =\frac{1}{\cos \rho}\left( {\rm d} \rho^2-\sin^2
 \rho\,{\rm d}\theta^2
 \right)} \\[8pt]
  \displaystyle{K =-\frac{\sin^2 \rho}{2\cos \rho} } &\quad
  \displaystyle{K =-\frac{\sin^2 \rho}{2\cos \rho} } \\[12pt]
 \mbox {$\bullet$ Euclidean space  ${\bf E}^2$}&\quad\mbox {$\bullet$
 Minkowskian spacetime ${\bf M}^{1+1}$}\\[4pt]
 z=0;\ (\lambda_1,\lambda_2)=(0,1)&\quad
 z=0;\ (\lambda_1,\lambda_2)=(0,{\rm i})\\[4pt]
  \displaystyle{{\rm d} s^2 =  {\rm d} \rho^2+ \rho^2{\rm d}\theta^2
  } &\quad
  \displaystyle{{\rm d} s^2 =  {\rm d} \rho^2- \rho^2{\rm d}\theta^2} \\[2pt]
  \displaystyle{K =0 } &\quad
  \displaystyle{K =0} \\[6pt]
 \mbox {$\bullet$ {Deformed hyperbolic space} ${\bf H}_z^2$}&\quad\mbox
 {$\bullet$ {Deformed de Sitter spacetime} ${\bf dS}^{1+1}_z$}\\[4pt]
 z=1;\ (\lambda_1,\lambda_2)=(1,1)&\quad
 z=1;\ (\lambda_1,\lambda_2)=(1,{\rm i})\\[4pt]
 \displaystyle{{\rm d} s^2 =\frac{1}{\cosh \rho}\left( {\rm d} \rho^2+\sinh^2
 \rho\,{\rm d}\theta^2
 \right)} &\quad
 \displaystyle{{\rm d} s^2 =\frac{1}{\cosh \rho}\left( {\rm d} \rho^2-\sinh^2
 \rho\,{\rm d}\theta^2
 \right)} \\[8pt]
 \displaystyle{K =-\frac{\sinh^2 \rho}{2\cosh \rho} } &\quad
  \displaystyle{K =-\frac{\sinh^2 \rho}{2\cosh \rho} } \\[8pt]
 \hline
 \end{array}
 $$
 \hfill}
 \end{table}

\medskip
\noindent Accordingly, the kinetic energy  (\ref{ca}) is
transformed into
\begin{equation*}
 {\cal T}^{\rm \rm I}_z(\rho,\theta;\dot \rho,\dot \theta)=\frac
 {1}{2\cosh(\lambda_1 \rho)}
 \left(\dot \rho^2  +\lambda_2^2\,\frac{\sinh^2(\lambda_1 \rho)}{\lambda_1^2} \,
 \dot \theta^2
 \right),
\end{equation*}
and  the free motion Hamiltonian (\ref{bg}) is written as
\begin{equation*}
 \widetilde{H}^{\rm I}_z=\frac 12 \cosh(\lambda_1
 \rho)\left(p_\rho^2 +\frac{\lambda_1^2}{\lambda_2^2\sinh^2(\lambda_1 \rho)} \,
 p_\theta^2\right),

\end{equation*}
where $\widetilde{H}^{\rm I}_z =2 {\cal H}^{\rm I}_z$. There is a
{unique} constant of the motion ${ C}^{(2)}_z\equiv{ C}_{z,(2)}$
(\ref{casdos}) which in terms of the new phase space is simply
given by
\begin{equation*}
  \widetilde{C}_z=p_\theta^2 ,
\end{equation*}
provided that $\widetilde{C}_z= 4\lambda_2^2  { C}^{(2)}_z$.
 This allows us to apply a   radial-symmetry reduction:
\begin{equation*}
 \widetilde{H}^{\rm I}_z =\frac 12 \cosh(\lambda_1
 \rho)\, p_\rho^2  +\frac{\lambda_1^2 \cosh(\lambda_1
 \rho)}{2\lambda_2^2\sinh^2(\lambda_1 \rho)} \, \widetilde{C}_z.

\end{equation*}
We remark that the explicit integration of the geodesic motion on
all these spaces can be explicitly performed in terms of elliptic
integrals.

\subsubsection{The superintegrable case}

A MS
  Hamiltonian is given by
\begin{equation*}
  {\cal H}^{\rm MS}_z =\frac 12  J_+^{(2)} {\rm e}^{ z J_-^{(2)}}=\frac12
\left( \frac {\sinh z
 q_1^2}{z q_1^2} \, {\rm e}^{z q_1^2}{\rm e}^{2z q_2^2}  p_1^2   +
 \frac {\sinh z q_2^2}{z q_2^2} \, {\rm e}^{ z q_2^2}  p_2^2  \right),
\end{equation*}
 since there exists  an additional (and functionally independent)
 constant of the motion~\cite{Deform}:
\begin{equation}
 {\cal I}_z=\frac {\sinh z
 q_1^2}{2 z q_1^2} \, {\rm e}^{z q_1^2}  p_1^2 .
 \label{bjjzz}
\end{equation}
 This choice corresponds to  the
 kinetic energy
\begin{equation*}
 {\cal T}^{\rm MS}_z(q_i,\dot q_i)=\frac 12 \left(\frac {z
 q_1^2}{\sinh z q_1^2} \, {\rm e}^{-z q_1^2}{\rm e}^{-2 z q_2^2} \,\dot
 q_1^2   +
 \frac {z q_2^2}{\sinh z q_2^2} \, {\rm e}^{-z q_2^2}\, \dot q_2^2
 \right)  ,
\end{equation*}
whose associated {metric} is
\begin{equation*}
 {\rm d} s_{\rm MS}^2=\frac {2z q_1^2}{\sinh z
 q_1^2} \, {\rm e}^{-z q_1^2}{\rm e}^{-2 z q_2^2} \,{\rm d} q_1^2   +
  \frac {2 z q_2^2}{\sinh z q_2^2} \, {\rm e}^{-z q_2^2} \, {\rm d} q_2^2 .
\end{equation*}
Surprisingly enough, the computation of the Gaussian curvature $K$
for ${\rm d} s_{\rm MS}^2$ gives that $K=z$. Therefore, we are dealing
with a space of constant curvature which is just  the deformation
parameter $z$. In \cite{plb} it was shown that a certain change of
coordinates (that includes the signature parameter $\lambda_2$)
transforms the metric into
 \begin{equation*}
 {\rm d} s_{\rm MS}^2=
  {\rm d} r^2  +\lambda_2^2\,\frac{\sin^2(\lambda_1 r)}{\lambda_1^2} \, {\rm d} \theta^2   ,
\end{equation*}
which exactly coincides with the metric of the Cayley--Klein
spaces written in geodesic polar coordinates
 $(r,\theta)$   provided that now $z=\lambda_1^2\equiv \kappa_1$ and
 $\lambda_2^2\equiv
 \kappa_2$.
Obviously, after this change of variables the geodesic motion can
be reduced to a ``radial'' 1D system:
\begin{equation*}
 \widetilde{H}^{\rm MS}_z=\frac 12 \, p_r^2
 +\frac{\lambda_1^2}{2\lambda_2^2\sin^2(\lambda_1 r)} \,
  \widetilde{C}_z  ,

\end{equation*}
where $\widetilde{H}^{\rm MS}_z=2{\cal H}^{\rm MS}_z$ and
$\widetilde{C}_z=p_\theta^2$ is, as in the previous case, the usual
generalized momentum for the $\theta$ coordinate.

\subsubsection{A more general case}

At this point, one could wonder whether there exist other choices
for the Hamiltonian yielding constant curvature. In fact, let us
consider the generic Hamiltonian (\ref{bbfff}) depending on $f$.
If we compute the
 general expression for the  2D Gaussian curvature in terms of the function
$f(x)$ we f\/ind that
 \begin{gather*}
K(x)=z\left(f^\prime(x)\cosh x  +\left(
 f^{\prime\prime}(x)-f(x)-\frac{{f^\prime}^2(x)}{f(x)}
 \right) \sinh x
 \right),
 \end{gather*}
 where $x\equiv zJ_-=z(q_1^2+q_2^2)$, $f^\prime=\frac{{\rm
 d}f(x)}{{\rm d}
 x}$ and $f^{\prime\prime}=\frac{{\rm d}^2f(x)}{{\rm d} x^2}$. Thus, in
general, we obtain spaces with variable curvature. In order to
characterize the constant curvature cases, we can def\/ine $g:=
f^\prime/f$ and write
\[
K/z = f^\prime \cosh x + \left(f^{\prime \prime} -f -
(f^\prime)^2/f \right) \sinh x  =  f\left( g \cosh x + (g^\prime
-1)\sinh x \right).
\]
If we now  require $K$ to be a constant we get the equation
\[
K^\prime=0 \equiv  2y\cosh x + y^\prime \sinh x = 0 , \qquad
\mbox{where}\quad y:=2g^\prime + g^2 -1 .
\]
The solution for this equation yields
\[
{{y=\frac{A}{\sinh^2 x}}},
\]
where $A$ is a constant, and solving for ${{g}}$, we  get for
${{F:=f^{\frac{1}{2}}}}$ the equation
\[
F^{\prime \prime}= \frac{1}{4} \left( 1+ {\frac{A}{\sinh
^2x}}\right)F ,
\]
whose  general solution is (${{A:=\lambda(\lambda-1)}}$):
\begin{equation*}
{{F= (\sinh x)^\lambda\left\{ C_1 \bigl(\sinh
(x/2)\bigr)^{(1-2\lambda)} + C_2\bigl( \cosh
(x/2)\bigr)^{(1-2\lambda)}\right\} }},
\end{equation*}
where $C_1$ and $C_2$ are two integration constants.

Therefore, many dif\/ferent solutions lead to 2D constant
curvature spaces. However,   we must impose as additional
condition that $\lim\limits_{x\to 0}{f}=1$. In this way   we
obtain that only the cases with $A=0$ are possible, that is,
either $\lambda=1$ or $\lambda=0$. Hence the two elementary
solutions   are just the Hamiltonians
\begin{equation*}
{\cal H}_z=\frac 12  J_+ {\rm e}^{\pm  z J_-},
\end{equation*}
and the curvature of their associated spaces is $K=\pm z$.

\subsection{3D   curved manifolds}

The study of the 3D case follows exactly the same pattern. The
three-particle symplectic realization of  $sl_z(2,\mathbb R)$
(with all $b_i=0$) is obtained from (\ref{zsymp}):
\begin{gather*}
J_-^{(3)} = q_1^2+q_2^2+q_3^2\equiv \>q^2  ,\nonumber\\
 J_+^{(3)} =
  \frac {\sinh z q_1^2}{z q_1^2}\,  p_1^2\, {\rm e}^{z q_2^2}{\rm
e}^{z q_3^2} +
 \frac {\sinh z q_2^2}{z q_2^2} \, p_2^2  \, {\rm e}^{-z q_1^2} {\rm
e}^{z q_3^2}
 +
 \frac {\sinh z q_3^2}{z q_3^2} \, p_3^2 \,   {\rm e}^{-z q_1^2} {\rm
e}^{-z q_2^2}
 \\
 J_3^{(3)} =
\frac {\sinh z q_1^2}{z q_1^2 }\,  q_1 p_1   {\rm e}^{z q_2^2}
{\rm e}^{z q_3^2} + \frac {\sinh z q_2^2}{z q_2^2 } \, q_2 p_2
{\rm e}^{-z q_1^2}{\rm e}^{z q_3^2}  + \frac {\sinh z q_3^2}{z
q_3^2 } \, q_3 p_3   {\rm e}^{-z q_1^2}{\rm e}^{-z q_2^2}
.\nonumber
\end{gather*}
By construction,  these generators  Poisson-commute with  the
three integrals $\{ { C}_z^{(2)},
 { C}_z^{(3)}\equiv C_{z,(3)}$, $C_{z,(2)}
\}$ given in (\ref{fk}):
\begin{gather}
  { C}_z^{(2)}    = \frac {\sinh z q_1^2 }{z q_1^2 } \,
\frac {\sinh z q_2^2}{z q_2^2} \left({q_1}{p_2} -
{q_2}{p_1}\right)^2 {\rm e}^{-z q_1^2}{\rm e}^{z
q_2^2} ,\nonumber\\
 C_{z,(2)} = \frac {\sinh z q_2^2 }{z q_2^2 } \, \frac
{\sinh z q_3^2}{z q_3^2} \left({q_2}{p_3} - {q_3}{p_2}\right)^2
{\rm e}^{-z q_2^2}{\rm e}^{z
q_3^2} ,\nonumber\\
 { C}_z^{(3)}   = \frac {\sinh z
q_1^2 }{z q_1^2 } \, \frac {\sinh z q_2^2}{z q_2^2}
\left({q_1}{p_2} - {q_2}{p_1}\right)^2 {\rm e}^{-z q_1^2}{\rm
e}^{z
q_2^2} {\rm e}^{2 z q_3^2} \label{3cas}\\
\phantom{{ C}_z^{(3)}   =}{}+ \frac {\sinh z q_1^2 }{z q_1^2 } \,
\frac {\sinh z q_3^2}{z q_3^2} \left({q_1}{p_3} -
{q_3}{p_1}\right)^2 {\rm e}^{-z q_1^2} {\rm e}^{  z
q_3^2}\nonumber  \\
\phantom{{ C}_z^{(3)}   =}{} + \frac {\sinh z q_2^2 }{z q_2^2 } \,
\frac {\sinh z q_3^2}{z q_3^2} \left({q_2}{p_3} -
{q_3}{p_2}\right)^2 {\rm e}^{-2z q_1^2}{\rm e}^{-z q_2^2} {\rm
e}^{z q_3^2}  . \nonumber
\end{gather}

\subsubsection{QMS free motion: non-constant curvature}

If we now consider the {kinetic energy} ${\cal T}_z(q_i,\dot q_i)$
coming from  the Hamiltonian
\begin{equation}
{\cal H}_z(q_i,p_i)=\frac 12 J_+^{(3)}, \label{otroham}
\end{equation}
it corresponds to  the free Lagrangian~\cite{Checz}
\begin{equation*}
{\cal T}_z= \frac 12 \left(\frac
 {z q_1^2}{\sinh z q_1^2} \, {\rm e}^{-z q_2^2}{\rm e}^{-z q_3^2} \dot
q_1^2   +
 \frac {z q_2^2}{\sinh z q_2^2} \, {\rm e}^{z q_1^2}{\rm e}^{-z q_3^2}
\dot q_2^2    +
 \frac {z q_3^2}{\sinh z q_3^2} \, {\rm e}^{z q_1^2}{\rm e}^{z q_2^2}
\dot q_3^2
 \right)  ,
\end{equation*}
 that def\/ines a {geodesic f\/low on a 3D
 Riemannian space} with    metric
\begin{equation*}
{\rm d} s^2=\frac {2z q_1^2}{\sinh z
 q_1^2} \, {\rm e}^{-z q_2^2}{\rm e}^{-z q_3^2} \,{\rm d} q_1^2   +
  \frac {2 z q_2^2}{\sinh z q_2^2} \, {\rm e}^{z q_1^2}{\rm e}^{-z
q_3^2}\, {\rm d} q_2^2   +
  \frac {2 z q_3^2}{\sinh z q_3^2} \, {\rm e}^{z q_1^2}{\rm e}^{z
q_2^2}\, {\rm d} q_3^2 .
\end{equation*}
The corresponding sectional curvatures $K_{ij}$ are
\begin{gather*}
K_{12}=\frac z4 \,{\rm e}^{-z \>q^2}\left( 1+ {\rm e}^{2 z q_3^2}-
2 {\rm e}^{2z
\>q^2}\right)  ,\\
K_{13}=\frac z4 \,{\rm e}^{-z \>q^2}\left( 2- {\rm e}^{2 z q_3^2}+
{\rm e}^{2 z q_2^2}{\rm e}^{2 z q_3^2}- 2 {\rm e}^{2z
\>q^2}\right)  ,\\
K_{23}=\frac z4 \,{\rm e}^{-z \>q^2}\left( 2-  {\rm e}^{2 z
q_2^2}{\rm e}^{2 z q_3^2}- 2 {\rm e}^{2z \>q^2}\right).
\end{gather*}
The following nice expression for the scalar curvature $K$ is
found:
\begin{equation*}
K_{12}+K_{13}+K_{23}=-\frac 52\,z\,\sinh(z\>q^2) = K/2.
\end{equation*}

Once again, the radial symmetry can be explicitly emphasized
through   new canonical coordinates $(\rho,\theta,\phi)$ def\/ined
by:
\begin{gather}
\cosh^2(\lambda_1\rho)= {\rm e}^{2z \>q^2} ,\nonumber\\
\sinh^2(\lambda_1\rho)\cos^2(\lambda_2\theta)={\rm e}^{2 z q_1^2}{\rm e}^{2 z
q_2^2}\left({\rm
e}^{2 z q_3^2}-1 \right)  ,\nonumber\\
 \sinh^2(\lambda_1\rho)\sin^2(\lambda_2\theta)\cos^2\phi={\rm e}^{2 z
q_1^2}
\left({\rm e}^{2 z q_2^2}-1 \right)  ,\nonumber\\
  \sinh^2(\lambda_1\rho)\sin^2(\lambda_2\theta)\sin^2\phi=  {\rm
e}^{2 z q_1^2}-1   , \label{xc}
\end{gather}
where $z=\lambda_1^2$ and $\lambda_2\ne 0$ is the additional signature
parameter, that will allow for  the presence of relativistic
spaces. Under this change of variables, the metric  is transformed
into
\begin{equation*}
 {\rm d} s^2=\frac {1}{\cosh(\lambda_1 \rho)}
 \left( {\rm d} \rho^2  +\lambda_2^2\,\frac{\sinh^2(\lambda_1 \rho)}{\lambda_1^2} \left(  {\rm d}
 \theta^2 + \frac{\sinh^2(\lambda_2 \theta)}{\lambda_2^2} \,{\rm d}\phi^2  \right) \right).

\end{equation*}
This is just the  metric of the 3D Riemannian and relativistic
spacetimes written in geodesic polar coordinates and multiplied by
a global factor ${1}/{\cosh(\lambda_1 \rho)}$ that encodes the
information concerning the variable curvature of the space.

Sectional and scalar curvatures are now written in the form
\begin{equation*}
K_{12}=K_{13}= -\frac 12 \lambda_1^2 \,\frac{\sinh^2(\lambda_1
\rho)}{\cosh(\lambda_1
 \rho)},\qquad K_{23}=K_{12}/2 ,\qquad
K= -\frac 52 \lambda_1^2 \,\frac{\sinh^2(\lambda_1 \rho)}{\cosh(\lambda_1
 \rho)} .
\end{equation*}

Therefore, according to the values of $(\lambda_1,\lambda_2)$ we have
obtained a deformation of the 3D sphere $(i,1)$, hyperbolic
$(1,1)$, de Sitter   $(1,i)$ and anti-de Sitter $(i,i)$ spaces.
The ``classical" limit $z\to 0$   corresponds to a
zero-curvature limit leading to the proper
  Euclidean $(0,1)$ and Minkowskian $(0,i)$ spaces.

The QMS  Hamiltonian (\ref{otroham}), that determines the free
motion on the above spaces, and its three integrals of the motion
(\ref{3cas}) are written in terms of the new canonical coordinates
$(\rho,\theta,\phi)$ and conjugated momenta $(p_\rho,p_\theta,p_\phi)$
as{\samepage
\begin{gather}
{\tilde H}_z=\frac 12 {\cosh(\lambda_1 \rho)}
 \left( p_\rho^2  + \frac{\lambda_1^2}{\lambda_2^2\sinh^2(\lambda_1 \rho)} \left(
 p_\theta^2 + \frac{\lambda_2^2}{\sin^2(\lambda_2 \theta)}\,  p_\phi^2  \right) \right),
\nonumber\\
{\tilde C}_z^{(2)}=p_\phi^2, \qquad {\tilde C}_z^{(3)}=p_\theta^2+
\frac{\lambda_2^2}{\sin^2(\lambda_2 \theta)}\, p_\phi^2, \qquad {\tilde
C}_{z,(2)}=\left(\cos\phi\, p_\theta-\lambda_2\frac{\sin\phi\,
p_\phi}{\tan(\lambda_2\theta)}  \right)^2 , \label{mascas}
\end{gather}
provided that ${\tilde H}_z= 2{\cal H}_z$, ${\tilde  C}_z^{(2)}=4{
C}_z^{(2)}$, ${\tilde  C}_{z,(2)}= 4 \lambda_2^2 { C}_{z,(2)}$ and
${\tilde  C}_z^{(3)}= 4 \lambda_2^2 { C}_z^{(3)}$.}

Furthermore the  set of three functions $\{{\tilde
C}_z^{(2)},{\tilde C}_z^{(3)},{\tilde H}_z\}$,   which
characterizes the  complete integrability of the Hamiltonian,
allows us to
 write  three equations, each of them depending on a canonical pair:
\begin{gather}
{\tilde  C}_z^{(2)}(\phi,p_\phi)=p_\phi^2  ,\qquad {\tilde
C}_z^{(3)}(\theta,p_\theta)=p_\theta^2+\frac{\lambda_2^2}{
\sin^2(\lambda_2\theta)}\,{\tilde  C}_z^{(2)}
,\nonumber\\
{\tilde H}_z(\rho,p_\rho)=\frac 12 {\cosh(\lambda_1 \rho)}
 \left(   p_\rho^2+
\frac{\lambda_1^2}{\lambda_2^2 \sinh^2(\lambda_1 \rho) } \,{\tilde  C}_z^{(3)}
\right). \label{fee}
\end{gather}
Therefore the Hamiltonian is separable and  reduced to a 1D radial
system.

\subsubsection{MS free motion: constant curvature}

The following choice for the Hamiltonian
\begin{equation*}
{\cal H}_z^{\rm MS}=\frac 12 J_+^{(3)} \,{\rm e}^{ z J_-^{(3)}},
\end{equation*}
yields a MS system  since it has four (functionally independent)
constants of motion, the three universal  integrals (\ref{3cas})
together with   ${\cal I}_z$ (\ref{bjjzz}). In fact, this the 3D
version of the Hamiltonian described in Section 4.1.2.

The associated {kinetic energy} is
\begin{equation*}
{\cal T}^{\rm MS}_z=\frac 12 \left(\frac
 {z q_1^2}{\sinh z q_1^2} \, {\rm e}^{z q_1^2}  \dot
q_1^2   +
 \frac {z q_2^2}{\sinh z q_2^2} \, {\rm e}^{2z q_1^2}{\rm e}^{z q_2^2}
\dot q_2^2    +
 \frac {z q_3^2}{\sinh z q_3^2} \, {\rm e}^{2z q_1^2}{\rm e}^{2z q_2^2}{\rm e}^{z
q_3^2} \dot q_3^2
 \right)  ,
\end{equation*}
and the underlying metric reads
\begin{equation}
{\rm d} s_{\rm MS}^2=\frac {2z q_1^2}{\sinh z
 q_1^2} \, {\rm e}^{z q_1^2}  \,{\rm d} q_1^2   +
  \frac {2 z q_2^2}{\sinh z q_2^2} \, {\rm e}^{2z q_1^2}{\rm e}^{z
q_2^2}\, {\rm d} q_2^2   +
  \frac {2 z q_3^2}{\sinh z q_3^2} \, {\rm e}^{2 z q_1^2}{\rm e}^{2 z
q_2^2}{\rm e}^{z q_3^2}\, {\rm d} q_3^2 .
 \label{yb}
\end{equation}
This space is again a {Riemannian} one with
 constant sectional and scalar curvatures given by
\begin{equation*}
K_{ij}=z ,\qquad K=6z .
\end{equation*}
Through an appropriate change of coordinates~\cite{Checz} we
f\/ind that  (\ref{yb}) is transformed into the 3D Cayley--Klein
metric written in terms of   geodesic polar coordinates
$(r,\theta,\phi)$:
\begin{equation*}
 {\rm d} s^2_{\rm MS}= {\rm d} r^2  +\lambda_2^2\,\frac{\sin^2(\lambda_1 r)}{\lambda_1^2} \left(  {\rm d}
 \theta^2 + \frac{\sin^2(\lambda_2 \theta)}{\lambda_2^2} \,{\rm d}\phi^2  \right)   .

\end{equation*}
Therefore, according to the values of  $(\lambda_1,\lambda_2)$, this
metric provides  the  3D
 sphere $(1,1)$,
Euclidean $(0,1)$, hyperbolic   $(i,1)$, anti-de Sitter $(1,i)$,
Minkowskian $(0,i)$, and de Sitter $(i,i)$ spaces.

Now the MS Hamiltonian, ${\tilde H}_z^{\rm MS}=2{\cal H}_z^{\rm
MS}$, is written as
\begin{equation*}
{\tilde H}_z^{\rm MS}=\frac 12
 \left( p_r^2  + \frac{\lambda_1^2}{\lambda_2^2\sin^2(\lambda_1 r)} \left(
 p_\theta^2 + \frac{\lambda_2^2}{\sin^2(\lambda_2 \theta)}\,  p_\phi^2  \right) \right),

\end{equation*}
and the four functionally independent integrals are given by
(\ref{mascas}) and
\begin{equation*}
{\tilde I}_z =\left(\lambda_2\sin(\lambda_2\theta)\sin\phi\,
p_r+\frac{\lambda_1\cos(\lambda_2\theta)\sin\phi}{\tan(\lambda_1 r)}\,p_\theta +
\frac{\lambda_1\lambda_2\cos \phi }{\tan(\lambda_1 r)\sin(\lambda_2\theta)}\,p_\phi
\right)^2 ,
\end{equation*}
where  ${\tilde I}_z =4\lambda_2^2{\cal I}_z$. The {two sets}
$\{{\tilde H}_z^{\rm MS}, {\tilde C}_z^{(2)}, {\tilde
C}_z^{(3)}\}$ and $\{{\tilde H}_z^{\rm MS}, {\tilde C}_{z,(2)},
{\tilde I}_z\}$ consist of three functions in involution.
Similarly to (\ref{fee}),    this Hamiltonian  is also separable
and can be reduced to a~1D system:
 \begin{equation*}
{\tilde H}_z^{\rm MS}(r,p_r)=\frac 12
 \left( p_r^2  + \frac{\lambda_1^2}{\lambda_2^2\sin^2(\lambda_1 r)} \,{\tilde C}_z^{(3)} \right).
\end{equation*}

\section[$N$D spaces with variable curvature]{$\boldsymbol{N}$D spaces with variable curvature}

The generalization to arbitrary dimension is obtained through the
same procedure, and the starting point is the QMS Hamiltonian for
the $N$D  geodesic motion that, in the simplest case, reads:
\begin{equation*}
 {\cal H}_z = \frac 12
 J_+^{(N)}=\frac 12
 \sum_{i=1}^N
  \frac {\sinh z q_i^2}{z q_i^2}\,  p_i^2    \exp{\left( - z
 \sum_{k=1}^{i-1}  q_k^2+  z \sum_{l=i+1}^N   q_l^2 \right)} .
\end{equation*}
The
 geometric characterization of the underlying $N$D curved
 spaces follows the same path as in the 2D and 3D cases described in the
previous sections.

If we write the above Hamiltonian as
\begin{equation*}
{ {\cal{H}}_z = \frac{1}{2} \sum_{i=1}^N s_z(q_i^2)\,p_i^2\exp
\left( z\sum_{   k=1; k  \neq i}^N  {\rm{sgn}}(k-i)q_k^2 \right)
},
\end{equation*}
where $s_z(q_i^2)=   {\sinh z q_i^2}/(z q_i^2)$ and ${\rm
sgn}(k-i)$ is the sign of the dif\/ference $k-i$, we get again a~free Lagrangian:
\begin{equation*}
{ {\cal{T}}_z = \frac{1}{2} \sum_{i=1}^N \frac{(\dot q_i)^2
\exp\left(-z\sum\limits_{   k=1; k  \neq i}^N
{\rm{sgn}}(k-i)q_k^2\right)}{s_z(q_i^2)}} ,
\end{equation*}
with the    corresponding (diagonal) metric given by
\begin{equation}
 {\rm d} s^2 = \sum_{i=1}^N g_{ii}(q) \, {\rm d} q_i^2 ,\qquad
 g_{ii}(q)=\frac{\exp\left(-z\sum\limits_{   k=1; k  \neq i}^N
{\rm{sgn}}(k-i)q_k^2\right)}{s_z(q_i^2)}. \label{mmetric}
\end{equation}

It turns out that the most suitable way to understand the nature
of the problem as well as to enforce separability is to consider
two sets of new coordinates:

\begin{itemize}\itemsep=0pt

 \item[$\bullet$] $N+1$ ``collective" variables~\cite{BRcluster} $({\xi _0},{\xi
_1},\dots,{\xi _N})$. They play a similar role to the ambient
coordinates arising when $N$D Riemannian spaces of constant
curvature are embedded within ${\mathbb R}^{N+1}$.

 \item[$\bullet$]  $N$ ``intrinsic" variables  $(\rho,\theta_2,\dots,\theta_N)$ which describe
the $N$D space itself. They   are the analogous to the geodesic
polar coordinates on $N$D Riemannian spaces of constant
curvature~\cite{VulpiLett,CRMVulpi}.

 \end{itemize}

The above coordinates are def\/ined in terms of the initial $q_i$
by:
\begin{gather}
{{\xi_0^2}} ={{ \cosh^2 (\lambda_1\rho):= \prod_{i=1}^N \exp( 2z q_i^2)}} ,\nonumber\\
{{\xi_k^2}}= {{ \sinh^2 (\lambda_1\rho)\prod_{j=2}^{k}\sin^2\theta_j
\cos^2 \theta_{k+1} := \prod_{i=1}^{N-k} \exp (2z
q_i^2)\left(\exp(2zq_{N-k+1}^2)
-1\right)}} ,\label{changen}\\
{{\xi_N^2}}={{\sinh^2(\lambda_1\rho)\prod_{j=2}^N\sin^2\theta_j
}}:=\exp(2zq_{1}^2) -1, \nonumber
\end{gather}
 where $z=\lambda_1^2$, $k=1,\dots,N-1$, and hereafter a product $\prod_j^k$ such that $j>k$
is assumed to be equal to 1. Notice also that for the sake of
simplicity we have not introduced the additional signature
parameter $\lambda_2$ (which would have been associated with $\theta_2$).
This def\/inition is the $N$D generalization of the   change of
coordinates (\ref{xc}) given in the 3D case with $\theta=\theta_2$,
$\phi=\theta_3$ and $\lambda_2=1$.

Clearly, the $N+1$ collective variables are not independent and
they fulf\/il a pseudosphere relation (of hyperbolic type):
\begin{equation*}
\xi _0^2-\sum_{k=1}^N \xi _k^2 = 1.
\end{equation*}

The geodesic f\/low in the canonical coordinates  ${{(\rho,
\theta)}}$ and momenta ${{(\rho,  p_\theta)}}$ is then given by
the Hamiltonian ${\tilde {H}}_z=2 {\cal{H}}_z$:
\begin{gather*}
{{ {\tilde{H}}_z  = \frac 12 \cosh(\lambda_1\rho) \left( p_{\rho}^2
+\frac{\lambda_1^2}{\sinh^2 (\lambda_1\rho)} \sum_{i=2}^N
\left(\prod_{j=2}^{i-1}\frac{1}{\sin^2
\theta_j}\right)p_{\theta_i}^2\right) }},
\end{gather*}
and the (left) integrals of the motion $\tilde C^{(m)}_z= 4
C^{(m)}_z$ are found to be
\begin{gather*}
\tilde C^{(m)}_z=\sum_{i=N-m+2}^N    \left(\prod_{j=N-m+2}^{i-1}
\frac{1}{\sin^2 \theta_j}\right)p_{\theta_i}^2,\qquad m=2,\dots,N
.
\end{gather*}
By taking into account the $N$ functions $\{{\tilde{H}}_z,\tilde
C^{(m)}_z\}$, we obtain the following set of $N$ equations, each
of them depending on a single canonical pair, which shows the
reduction of the system to  a 1D problem:
\begin{gather*}
\tilde C^{(2)}_z(\theta_N,p_{\theta_N})=p_{\theta_N}^2,\nonumber\\
\tilde
C^{(m)}_z(\theta_{N-m+2},p_{\theta_{N-m+2}})=p_{\theta_{N-m+2}}^2+\frac{
C^{(m-1)}_z}{\sin^2\theta_{N-m+2}},\qquad m=3,\dots,N,\nonumber\\
{\tilde{H}}_z(\rho,p_\rho) = \frac 12 \cosh(\lambda_1\rho) \left(
p_{\rho}^2 +\frac{\lambda_1^2}{\sinh^2 (\lambda_1\rho)}  \,\tilde
C^{(N)}_z  \right).
\end{gather*}
We stress that these models can be extended to incorporate
appropriate  interactions with an external central f\/ield,
preserving superintegrability. This will be achieved by modifying
the Hamiltonian by adding an arbitrary function of $J_-$, as we
shall see in the next Section.

Finally we remark that the corresponding generalization to $N$D
spaces with constant curvature can be obtained by considering the
MS Hamiltonian
\begin{equation}
 {\cal H}^{\rm MS}_z = \frac 12 \,
 J_+^{(N)}\,{\rm e}^{ z J_-^{(N)}}  =  \frac 12\,{\rm
e}^{ z \>q^2}
 \sum_{i=1}^N
  \frac {\sinh z q_i^2}{z q_i^2}\,  p_i^2    \exp{\left( - z
 \sum_{k=1}^{i-1}  q_k^2+  z \sum_{l=i+1}^N   q_l^2 \right)} .
\label{msham}
\end{equation}
Under a suitable change of coordinates, similar to (\ref{changen})
but involving  a dif\/ferent radial coordinate~$r$ instead of
$\rho$, this Hamiltonian leads to the MS geodesic motion on ${\>
S}^N$, ${\> H}^N$ and ${\> E}^N$ in the proper geodesic polar
coordinates which can be found in~\cite{VulpiLett,CRMVulpi}.

\section{QMS potentials}

As we have just noticed, we can also consider more general $N$D
QMS Hamiltonians based on $sl_z(2,{\mathbb R})$ (\ref{zsymp}) by
considering arbitrary $b_i$'s (contained in $J_+$) and adding some
functions depending on $J_-$; hereafter we drop the index ``$(N)$"
in the generators. The family of Hamiltonians that we consider has
the form (see \cite{jpa2D} for the 2D construction):
\begin{equation*}
{\cal H}_z=\frac 12 J_+ \, f (zJ_-
 )+{\cal U} (zJ_- )  ,
\end{equation*}
where the arbitrary smooth functions $f$ and $\cal U$ are such
that
\begin{equation*}
\lim_{z\to 0}{\cal U}(zJ_-)={\cal V}(J_-), \qquad \lim_{z\to
0}f(zJ_-)=1.
\end{equation*}
This, in turn, means that
\begin{equation*}
\lim_{z\to 0} {\cal H}_z= \frac 12 \,\>p^2 + {\cal V}(\>q^2)
+\sum_{i=1}^N\frac{b_i}{2q_i^2},
\end{equation*}
recovering the superposition of a central potential ${\cal
V}(J_-)\equiv {\cal V}(\>q^2)$ with $N$ centrifugal terms on~${\>
E}^N$~\cite{Evansa}. Such a ``f\/lat" system has a (non-deformed)
$sl(2,{\mathbb R})$ coalgebra symmetry as given in Theorem 1.

We recall that the function $f (zJ_-
 )$ gives us the type of
{curved background}, which is characterized by the metric  ${\rm d}
s^2/f (z\>q^2 )$ where ${\rm d} s^2$ is the variable curvature metric
associated to ${\cal H}_z=\frac 12 J_+$ and given in
(\ref{mmetric}). The two special cases with $f (zJ_-
 )={\rm e}^{\pm z J_-}$ give rise to Riemannian spaces of constant sectional
curvatures, all equal to $\pm z$ (as (\ref{msham})).

In particular,  QMS deformations of the
 $N$D  SW
system would be given by any  ${\cal U}$ such that
\begin{equation*}
\lim_{z\to 0} {\cal U}(zJ_-) = \omegaJ_- ,
\end{equation*}
and for the $N$D generalized KC potential we can consider  ${\cal U}$
functions such that
\begin{equation*}
\lim_{z\to 0} {\cal U}(zJ_-)= -k/\sqrt{J_-}.
\end{equation*}
In both cases   centrifugal type potentials come from the $b_i$'s
terms contained in $J_+f (zJ_-  )$.

With account of the geometrical arguments, the following QMS SW
system on  spaces with non-constant curvature (\ref{mmetric}) has
been proposed in \cite{jpa2D}:
\begin{equation*}
{\cal H}^{\rm SW}_z=\frac 12 J_+ +
 {\omega} \,\frac{\sinh{z J_-}}{z},
\end{equation*}
while a   candidate for a generalized {KC system on such spaces}
is given by the formula \cite{jpa2D}:
\begin{equation*}
{\cal H}^{\rm KC}_z=\frac 12 J_+ -
 k \,\sqrt{  \frac{2z}{{\rm e}^{2 z J_-}-1}} \,{\rm e}^{2 z J_-} .
\end{equation*}

In the constant curvature case, the approach here presented allows
us to recover the known results for the SW potential on Riemannian
spaces with constant curvature, as well as their generalization to
relativistic spaces (whenever the signature parameter $\lambda_2$ is
considered).

In particular,  the MS {SW system on $N$D spaces of constant
curvature} is given by the Hamiltonian
\begin{equation*}
 {\cal H}^{\rm MS,SW}_z=\frac 12 J_+ {\rm e}^{ z J_-}+
 {\omega} \,\frac{\sinh{z J_-}}{z} \,{\rm e}^{ z J_-}\equiv {\cal H}^{\rm
SW}_z   {\rm e}^{ z J_-} ,
\end{equation*}
and the additional constant of the motion that provides the MS
property reads
\begin{equation*}
{\cal I}_z=\frac {\sinh z q_1^2}{2 z q_1^2} \, {\rm e}^{z q_1^2}
p_1^2 +\frac{z b_1}{2\sinh z q_1^2} \, {\rm e}^{z
q_1^2}+\frac{\omega}{2z}\,{\rm e}^{2 z q_1^2} .
\end{equation*}

The corresponding results for the generalized KC system are
currently under investigation.

\section{Concluding remarks}

The main message that we would like to convey to the scientif\/ic
community through the present paper is that  ``Superintegrable
Systems are not rare!". Indeed, in our approach they turn out to
be a natural manifestation of coalgebra symmetry: as such, they
can be equally well constructed on a f\/lat or on a curved
background, the latter being possibly equipped with a variable
curvature. Moreover, and, we would say,  quite remarkably the
construction holds for an arbitrary number of dimensions.

In that perspective, the most interesting  problems that are still
open are in our opinion the following ones:

\begin{enumerate}\itemsep=0pt
\item The explicit integration of  the equations of motion for (at
least some) of the prototype examples we have introduced in the
previous sections; \item The construction of the
quantum-mechanical counterpart of  our approach.

\end{enumerate}

\noindent As for the former point, partial  results have already
been obtained, and a detailed description of the most relevant
examples will be published soon. The latter point, in particular
as far as  the non-standard deformation of $sl(2,\mathbb R)$ is
concerned, is however  more subtle and deserves careful
investigation (which is actually in progress). In fact, f\/irst of
all  one has to f\/ind a proper $\infty$-dimensional
representation of such a non-standard deformation in terms of
linear operators acting on a suitably def\/ined Hilbert space,
ensuring self-adjointness of the Hamiltonians; second, and
certainly equally important, at least in some physically
interesting special cases one would like to exhibit the explicit
solution of the corresponding spectral problem.

\subsection*{Acknowledgements}

This work was partially supported  by the Ministerio de
Educaci\'on y Ciencia   (Spain, Project FIS2004-07913),  by the
Junta de Castilla y Le\'on   (Spain, Project VA013C05), and by the
INFN--CICyT (Italy--Spain).

\pdfbookmark[1]{References}{ref}

redpajama_set_name: RedPajamaArXiv


As my Hayter rough cut mower became difficult to start, I decided on a de-coke, and noticed that the exhaust valve tappet clearance was nil! after doing the de-coke and fixing the valves, the re-assembly was quite straight foreword EXCEPT I cant remember how those fiddly throttle springs go back! can someone who knows this engine layout help please? I have painted the holes white so you can see them in the photo, and starting at the top, I think that is the main throttle cable hole, and the bottom one is the governor lever, BUT there are THREE springs, a long one, and two together (one inside the other) just cant remember what hole went where! if you have a photo of the same engine with the springs on... it would be a great help!!
Hope that this helps. On this application the light spring (anti surge/rattle)wrapped around the governor link hooks into the same holes as the link but on others there will be a small hole adjacent to the link ones.
Perhaps a case for using the digi camera before as well as after!
Thanks for that, My phone packed up some weeks ago, and I have a cheap "Throwaway phone" that does not have a camera, so that is why I did not take a photo first, but I did stare at it for several seconds, so that I would remember the layout.... But I dont!...Bugger!
Dare I say there is still good old paper and pencil!!!
I would agree with that. I recently pulled an Atco Standard apart and took digi pics as well as a fairly rough non-artistic sketch. When it came to reassembling the rods between the frames with their multiple spacers and collars the drawing was of more use than the images! Perhaps there's a place for both?
Out there , someone has a Hayter with the B&S 5 HP engine on it, a photo would be nice of the spring layout, I shall try Wristpins diagram when I get down the barn, and hope that works.


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text:
Rikers Island Spared by Irene, But What About Next Time?
By Rebecca McCray, Staff Reporter, American Civil Liberties Union
Criminal Law Reform
Cruel, Inhuman, and Degrading Conditions
Last month, I wrote about the invisibility of Rikers Island on some of New York City's subway maps. Unfortunately, this weekend Hurricane Irene taught us that it's not just a map on a train that fails to acknowledge the thousands of people on the Island.
In a press conference Friday afternoon, Mayor Michael Bloomberg announced that, despite the impressive array of hurricane evacuations and preparatory plans implemented to protect New Yorkers in other low-lying areas, Rikers Island would not be evacuated. Luckily for all of us, Irene's sweep through New York was much less disastrous than anticipated. Rikers Island was spared, but we learned something disturbing: the Department of Corrections does not have a large scale evacuation plan in place for the 12,000-plus prisoners of Rikers. Understandably, this revelation caused a flurry of backlash from concerned advocates.
It may seem extreme to plan for such an emergency, but after experiencing an earthquake and a hurricane in the same week, isn't it safe to say that we should be prepared for anything? New York City need not replicate the horror stories of those abandoned at Orleans Parish Prison after Hurricane Katrina.
Mayor Bloomberg and the Department of Corrections should take responsibility for the more than 12,000 lives at stake on Rikers Island. It is the city's duty to be prepared for the worst, protecting all its citizens in the event of a disaster — including those that live behind bars.
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Fight for everyone's rights - support the ACLU.
I Quit My Job to Protest My Company's Work on Building Killer Robots
A Secret CBP Team is Targeting and Detaining Innocent Travelers. We're Suing.
CBP Can't Detain Domestic Flight Passengers for Refusing Suspicionless ID Checks
Trump's War Pardons Are Sabotaging the Military Justice System
It's Past Time Congress Reined in the President's Emergency Powers

redpajama_set_name: RedPajamaCommonCrawl


Last weekend envelope users contributed 75% of our income.
As we begin a New Year it is an excellent time to stop and reflect on who we are as a Faith Community and where we would like to go in the near and distant future.
As most of you know, we have been doing a lot of footwork with respect to the possibility of expanding our church building. The reasons for expansion have been expressed with little or no contradiction. We have proceeded to meet with the diocesan realtor commission and we have further proceeded to seek permission from the bishop for furthering the process. All of this has been completed.
The year 2007 not only ushers in a new calendar year, it also brings us to a place where we must take a long and serious look at our parish life. We must not be afraid to ask ourselves some difficult questions. Expanding our building necessitates a large sum of money. Before embarking on this expensive venture, it would be financially and pastorally irresponsible of the pastor and the pastoral staff to avoid the task of looking at ourselves (the parish) in the mirror.
leaders we believe they must be presented to you, the parishioners. Some time will also be given for assessment of the situation.
Question for Adults: What areas of my life need to be visited by Jesus' light? We all have areas of our life that seem to be hidden in the dark recesses of our inner self. Sometimes they are so well hidden that we have difficulty seeing them or perhaps even realizing that they exist.
The visit of the Magi is significant in our lives because it tells us that Christ brought light to everyone, even to people who were not necessarily aware of the fact that they needed him.
Also, it is not a coincidence that one of the main characters in this story is King Herod, a man who wanted nothing to do with the new born child. King Herod represents all of us who want our own way in life and fear that the presence of Christ in our lives will destroy our comfortable practices. The presence of Herod in the story tells us that we must be open to the light if we are to benefit from it.
Hopefully, we are closer to the Magi than to Herod. But even the Magi needed to be touched by grace. Scripture does not elaborate but we must not fail to reflect. Did the visit to the Child change their lives or was it simply a one time visit that caused some excitement for these men? Did they return to their homes only to fall back into their old ways?
Let us take some time this week to dust off some of those dark corners of our self that need a little shine. Let's not allow the Christmas season to be dismissed and forgotten until next December.
Question for Children? Does the presence of Jesus in my life make a difference to me and to the people that live or play with me? How?
Take a few minutes to brainstorm a list of people who could be affected by the behavior of your child. Then, help them to think of ways that would help others be happy. Sharing is always a good way but try to encourage them to realize that they can share their joy and happiness with a smile and a cheerful word. A pleasant attitude does wonders for others in our company. As adults we are usually aware of this but children are not always aware. How does knowing that Jesus loves them make a difference in their attitude?
The Religious Vocation Commission of the Diocese of Manchester is offering evenings of camaraderie and discernment for women, ages 16-45, who are considering religious life. Our topic will be "Discernment." Have you wondered how to come to a decision about your vocation? This session will consist of a discussion of the steps of discernment leading to a decision with help of God. Join us at our next Naomi supper from 6:00-9:00p.m. Joseph House, 279 Cartier Street, Manchester. Please RSVP to Jude Christopher, Office of Vocations, 669-3100 ext. 153 or e-mail jchristopher@rcbm.org by January 8, 2007.


Do You Want a New Body?
Thanks to your body's amazing ability to continuously regenerate, many of your "parts" are far younger than that age on your birth certificate.
A natural function of your body is create new cells to replace those that have worn out. Your body generates new blood cells, new skin cells, new hair, and new cells for your digestive organs. In fact, most every part of your body is being replaced to some extent, each and every day.
Did you know that the muscles in your legs and the tissue within your gastrointestinal system are only about 15 years old? The red blood cells that deliver oxygen to every cell in your body are only four months old, on average. And the cells on the surface of your skin have a lifespan of just two weeks.
If our bodies are constantly rebuilding ourselves, then we all have the chance to greatly improve our bodies and, by extension, our health – starting right now.
While cell turnover naturally slows with age, giving your internal "mechanic" the right parts for the job (healthy food) and staying away from age-robbers that slow regeneration (such as too much fat, sugar and calories; smoking; too much alcohol: and excess stress) will make all the difference.
As it turns out your body has different regeneration modes. You can think of it like "slow mechanics" and "young mechanics". You can choose which body-repair mechanic will do the work, experts now believe. The key? Exercise.
Without physical activity, experts now suspect your body believes that it is winter – literally. Remember that you genetic coding isn't based on life as it is today, but as it was lived many thousands of years ago. And if you were sitting around day after day back then, it usually meant it was the cold-weather season, with you and your family huddled together inside for warmth, long past the season for running around gathering berries and hunting wild game. This also meant that your body needed to shift into hibernation mode; in fact, your top priority would be to burn as few calories as possible, so the slow mechanic took over, doing minimal work and allowing your bones too thin, our muscles to weaken, and more.
If you get up and move around every day, your genetic code says "Aha! I need stronger bones and muscles, more brain cells to figure out how to hunt that wild boar, and a stronger cardiovascular system to keep it all supplied with oxygen and nutrients while I forage for nuts and berries in the woods." Then the young mechanic gets to work, bolstering key body systems and creating strong new cells. Basically, you choose how healthy your body will stay.
Your body has a vast capacity for repair – and you don't have to be a scientist in a lab to experience the benefits. In my next blog I will explore 4 strategies for Maximum regeneration of your body.
All this is great news for those of us who want to turn our bodies around for the better. As you can see, it is totally achievable!


No Frizz Shampoo is ideal for all frizz-prone hair types. It is sulfate-free and silicone-free. Safe and formulated for colour and chemically treated hair.
Apply No Frizz Shampoo to wet hair. Massage into a rich lather, focusing on roots. Rinse. Follow with No Frizz Conditioner.
How is it different from other Living Proof Shampoos?
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What is the fragrance in No Frizz Shampoo?
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World Ozone Layer Day: go get chlorofluorocarbons and hydrochlorofluorocarbons!
Thanks to the entry into force of the Montreal Protocol, the World Ozone Layer Day has been held since September 16 every year. A day of awareness that, like the international treaty it commemorates, aims to protect this layer of the stratosphere. And in this, chlorofluorocarbons and hydrofluorocarbons play a fundamental role.
Nine years after the entry into force of the Montreal Protocol, the General Assembly of the United Nations proclaimed in its resolution 49/114 September 16th as the International Day for the Preservation of the Ozone Layer. From that moment, this day has been used to celebrate the enormous benefits that the protocol has for the planet.
"The initial goal of the Montreal protocol was the elimination of so-called chlorofluorocarbons. Now the Kigali amendment goes further and extends it to hydrofluorocarbons.
Since its universal ratification, the protocol has been highly successful in its initial objective of eliminating the production and use of chlorofluorocarbon compounds (CFCs) and other substances that destroy the ozone layer, achieving a significant reduction.
However, it is necessary to maintain a constant attention to the evolution of the emissions of these components. Without going any further, the Scientific Assessment Panel of the Montreal Protocol has recently published findings that show that there has been a setback in the eradication of CFC-11 gas, and the available data suggest that the problem is located in East Asia.
It is imperative to take decisive action to prevent any illegal consumption or production of this type, so that the Scientific Evaluation Panel and the Technology Assessment Group are preparing detailed reports on atmospheric monitoring of CFC-11, its dispersion model and sources specific issues, as well as possible pre-existing deposits.
The next follow-up milestone will be the Thirtieth Meeting of the Parties to the Montreal Protocol (MOP30), to be held during November 2018. In addition, this meeting will serve to encourage countries to ratify the Kigali Amendment, with a view to phasing out hydrofluorocarbons (HFCs).
"The celebration of the day of the ozone layer seeks to contribute to the preservation of this layer of the stratosphere and to build a sustainable future for life on our planet.
These compounds were initially promoted by the Montreal Protocol as an alternative to other substances that depleted the ozone layer. However, it was later shown that these substitutes had a high global warming potential, so they should no longer be considered as a viable option.
Through the program to reduce production and consumption of HFCs of the amendment, some 80,000 million tons of CO2 equivalent will be released by 2050, thus avoiding a temperature increase of approximately 0.5ºC between now and 2100. In this way, the amendment will contribute fully to the implementation of the Paris Agreement.
We are clearly at a critical moment in the struggle to restore the ozone layer. We must all be aware that the war is not yet won and that we will not achieve a happy ending unless all countries join forces and work towards the same goal, which should be no other than to work for a sustainable future for life in our planet.
Pablo Collazo Sánchez, responsible for environmental sustainability at Abengoa.


Appointments to CBJ Boards are considered by the Assembly Human Resources Committee (HRC) at their monthly meeting held the same Monday as the Assembly meetings at 6:00p.m. in the Assembly Chambers with the exception that no HRC meetings are held in October due to Assembly reorganization following the annual election.
Contact Deputy Municipal Clerk Diane Cathcart at 586-5278 or via email at City.Clerk@juneau.org.


The Sacrament of Confirmation is the final Sacrament of Initiation through which the baptized receive the gift of the Holy Spirit.
In the Diocese of Brooklyn candidates for the Sacrament must be at least thirteen years of age and have completed the necessary preparation through study and prayer.
Children seeking the Sacrament of Confirmation must be registered in our religious education program.
Adult Catholics who have not yet received the Sacrament should inquire about the process for completing their initiation in the RCIA program.


text:
Scyliorhinus haeckelii és una espècie de peix de la família dels esciliorrínids i de l'ordre dels carcariniformes.

Morfologia
 Els mascles poden assolir 60 cm de longitud total.

Hàbitat
És un peix marí que viu entre 37-402 m de fondària.

Distribució geogràfica
Es troba a l'oest de Veneçuela, Surinam, Brasil i l'Uruguai.

Referències

Bibliografia
 Compagno, L.J.V.: FAO species catalogue. Volum 4. Sharks of the world. An annotated and illustrated catalogue of shark species known to date. Part 2 - Carcharhiniformes. FAO Fish. Synop. 125(4/2):251-655. Any 1984.
 Compagno, L.J.V. i V.H. Niem, 1998. Scyliorhinidae. Catsharks. p. 1279-1292. A: K.E. Carpenter i V.H. Niem (eds.) FAO Identification Guide for Fishery Purposes. The Living Marine Resources of the Western Central Pacific. FAO, Roma, Itàlia.
 Compagno, L.J.V., D. Dando i S. Fowler, 2005. A field guide to the sharks of the world. Harper Collins Publishing Ltd., Londres, Gran Bretanya, 368 p.
 Coppola, S.R., W. Fischer, L. Garibaldi, N. Scialabba i K.E. Carpenter, 1994 SPECIESDAB: Global species database for fishery purposes. User's manual. FAO Computerized Information Series (Fisheries). Núm. 9. Roma, Itàlia, FAO. 103 p.
 Dulvy, N.K. i J.D. Reynolds, 1997. Evolutionary transitions among egg-laying, live-bearing and maternal inputs in sharks and rays. Proc. R. Soc. Lond., Ser. B: Biol. Sci. 264:1309-1315.

 Fritzsch, B. i P. Moller, 1995. A history of electroreception. p. 39-55. A P. Moller (ed.) Electric fishes: history and behavior. Fish and Fisheries Series 17. Chapman & Hall, Londres (Regne Unit).

Enllaços externs
 AQUATAB

haeckelii

redpajama_set_name: RedPajamaWikipedia


text:
HomePairingAmerican Chardonnay
American Chardonnay
The Kleenex of White Wine: American Chardonnay
We all have a story to tell about our introduction and seduction into the world of wine. Most of us started with white wines, unless Jewish and drinking Mogen David for Passover dinners. Many start with sweeter wines, but graduate rapidly to Chardonnay, when our palates mature in appreciation.
Chardonnay today holds court as the defacto American white wine. Many people do not even say, "I'll have a white wine", but instead say, "Give me a Chardonnay."
Chardonnays boast an impressive range of flavors from the expected buttered, oak overtones to the fresh, fruit flavors of apple, pear, tropical, citrus and melon, leaving a lasting palate impression. Chardonnay will pair well with poultry dishes, pork, seafood or recipes that have a heavy cream or butter base. Un-oaked Chardonnay goes well with guacamole, garlic, salads, grilled shrimp or even mild curry dishes.
Chardonnay is White Burgundy
With a long and distinguished following, haling from the French White Burgundies and Chablis, Chardonnay enjoys a very versatile image, with vintners offering a broad range of styles and structures. From rich, buttery Chardonnays that boast unctuous presence to the unoaked Chardonnays that allow the varietal character of the fruit and expression to be in the spotlight, this white wine is capable of accommodating most palates and just as many food pairing combinations.
Chardonnay is one of the most widely-planted grape varieties, second only to Airén among white wine grapes and planted in more wine regions than any other grape. It has a wide-ranging reputation for relative ease of cultivation and ability to adapt to different conditions. The grape is very "malleable", in that it reflects and takes on the impression of its terroir and winemaker.
As the leading white wine in the world, it has earned a comfortable and distinctive home in the United States for its distinctive flavor, full mouth feel, and compatibility with many dishes.


Re: What does mean by "Thread Name" and "Sample Time" columns?
Subject Re: What does mean by "Thread Name" and "Sample Time" columns?
results table. I doubt I can help.
name, then sample time (this is the default order of the columns).
> Thanks again, but I still not getting. I got following result in table.
> And my question is- For Thread Name- '1-287' there is Sample Time=47902 ms.
> So, is this Sample Time for '287th' user or '1 to 287' users collectively?
> Please add some focus, I am too confused.
> Thanks in advance !!
> > > Thank you Adrian for reply. I will keep your suggestion in mind.
> > > automatically ? Could you please elaborate this working, please?
> > behaviour, logging an exception encountered).
> > > > few virtual-users.
> > > > it shows the exact response time in miliseconds.
> > > > > Is someone over here to help me in my query?
> > > > > > I am new to JMeter. I ran simple test load test with 300 users.
> > > > > > TIME(ms) columns the following columns.
> > > > > > does this mean?
> > > > > > 1. What is sample time?
> > > > > > Therefore, I need your help !!


100% of your donations go to this ministry to educate, train others in discipleship and how to share their faith. In addition, we use monies to feed the homeless and those who do not have the means to support themselves, we call them those of meager means. There are NO administrative costs or salaries deducted from your generous sacrificial giving.
We have 18 volunteers, 3 of which are homeless themselves. No one is paid it is completely volunteer staff. We provide clothing for the homeless as well. None of the clothing is purchased from your money. All clothing is donated by other churches or by civic groups. Your ministry dollars go far.
We also ask you to not donate any monies that are due the local church by the way of tithe. We feel strongly about this. We fully believe your tithe belongs to God and should be administered at the local church level.
To donate please click on Paypal or you can mail a check made out to 2nd Chance Discipleship Ministries to 10583 N Government Way Hayden, Idaho 83835. All donations are tax deductible and at the end of the year you will be provided a letter verifying your contributions for your income taxes. God Bless you for your generosity.


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Army Recruitment Rally Slated for February at Bihar
Army Recruitment Drive At Likabali For Arunachal Pradesh Candidates
Indian Army Invites Engineering Graduates for Technical Graduate Course
Army Recruitment Rally Slated for Two Uttarakhand Districts in December
Indian Army to Protest against Soldiers' Killings at LOC
Indian Army to Host Recruitment Rally at Masimpur
Army Recruitment Drive Slated for Arunachal Pradesh at Tenga Valley
Indian Army Recruitment Rally Slated for Assam Candidates at Tezpur
Indian Army Invites Applicants from West Bengal
Indian Army Considers Reducing Troops in the Next Five Years
Home Army 3 Kumaon Rifles Regiment Celebrates 100th Raising Day
3 Kumaon Rifles Regiment Celebrates 100th Raising Day
AravindOct 23, 2017Army0
3 Kumaon Rifles regiment, which is the first Kumaoni battalion of British Indian Army and one of the oldest Indian Army regiments, recently celebrated their 100th raising day. Officials who conducted the event had earlier announced that more than 600 veterans would take part in the celebration. The officials had also stated that 40 "veer naris" (women who lost their husbands while they were serving the country) and 80 retired officers would also join the celebrations at Pithoragarh. The 3 Kumaon Rifles regiment is currently posted at Pithoragarh.
This battalion of the Indian Army has served the country has taken part in several successful operations in both post and pre-independence wars. Apart from that, the 3 Kumaon Rifles regiment were also deployed in many of the missions of the United Nation.
Kumaon Rifles Regiment
One of the serving officers from the 3 Kumaon Rifles regiment of Indian Army said, "During World War I, there was large-scale recruitment to meet the growing demand for troops. The reviewed recruitment policy saw a large number of Kumaonis recruited in Garhwal Rifles and other regiments, but they yearned to fight under their own flag. After a long struggle, this regiment was eventually raised."
The officer added, "The Kumaoni people, especially in the Champawat district, rose in rebellion against the British in 1857 under the leadership of Kalu Singh Mahara. Such was the impact of the rebellion that the British abandoned recruitment from the Kumaon region after 1857. However, since soldering for Kumaonis was not only an honourable profession but also a pecuniary compulsion, they covertly kept on seeking recruitment by changing their identity and joining Gurkha and Garhwali regiments."
When the battalion was initially raised it was christened as 4/39 Garhwal Rifles. However, the battalion was re-designated 4/39 Kumaon Rifles in a single month and eventually to 1/50 Kumaon Rifles. This unit was actually a part of the famous Battle of Sharon during the First World War.
One of the major of the regiment said, "It was during this battle that Kumaoni valour triumphed, under their own flag, for the first time in the field. The unit was renamed as 1st Kumaon Rifles on March 15, 1922 and later on as 3rd Battalion (Rifles) of the Kumaon Regiment. We are also the only Kumaoni regiment, which was not disbanded by the British. We are very proud of our service to the nation in the last 100 years."
TAGArmy Recruitment Indian Army
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text:
What is Nortriptyline 25mg Used For?
By: Health Community Member
Nortriptyline 25mg is used to treat mental or mood problems such as depression. It is being marketed under the brand names Pamelor and Aventyl. This medication is a tricyclic antidepressant, specifically of dibenzocycloheptene type, and is the active metabolite of the tertiary amine—amitriptyline. Tricyclic antidepressants work by inhibiting the reabsorption of the biogenic amine neurotransmitters: norepinephrine (NE) and serotonin (5HT). Tertiary amines inhibit the reuptake of both neurotransmitters, but secondary amines like nortriptyline are more selective of inhibiting NE over 5HT.
Tricyclic antidepressants (TCA) are rarely used as first-line or even second-line treatment for depression because of its extensive side effect profile and high overdose risk. However, it is still one of the most effective pharmacological therapies for severe depression or major depressive disorder of melancholic type, demonstrating greater efficacy over selective serotonin reuptake inhibitors (SSRIs). Your doctor may prescribe this drug for other reasons. Nortriptyline is being used off-label for treating various conditions such as premenstrual dysphoric disorder, bedwetting, social anxiety disorder, insomnia, migraine, and neuropathic pains, and as an aid for smoking cessation.
TCAs are used as third- or fourth-line treatment for major depressive disorder (MDD). The recommended dose for nortriptyline is 25–50mg per day in adult patients with MDD, with a maximum dose of 150mg. It can be taken in divided doses or once at bedtime for those who are bothered of its sedative effects. Adolescent and elderly patients must take 10–25mg once at bedtime, not exceeding 50mg per day.
Nortriptyline for depressed children and adolescents
Tricyclic antidepressants are contraindicated in depressed children. There are limited data supporting the use of nortriptyline and other TCAs in the treatment of depression in adolescents. In the early 1960s, tricyclic drugs were first used to treat depression in children and adolescents, but they were more commonly prescribed for treating nocturnal enuresis or bedwetting in children. According to a review of 14 trials testing the effectiveness of TCA against placebo:
Based on nine trials with 454 participants, no significant evidence that TCAs lead to higher rates of remission than placebo.
Based on 13 trials with 533 participants, there was evidence that MDD patients treated with a tricyclic agent had lower severity score compared to placebo, but the difference was small.
Based on six trials with 239 adolescent participants and two trials with 77 children participants, there was no evidence of differential rates of remission or response between the age groups.
Based on eight trials involving 414 adolescent participants, there was notable lowering of depression scores.
Based on three trials involving 64 children participants, there was no lowering of depression scores.
Review conclusion
The authors concluded that tricyclic drugs, such as amitriptyline, nortriptyline, desipramine, and imipramine are not useful for treating depression in prepubertal children. The use of TCAs for adolescent depression will likely provide modest effects. Prescribing a tricyclic as a second-line treatment for depressed adolescents who failed to respond to other pharmacotherapy or psychotherapy is also unwarranted, as one trial who used TCAs in this context also failed to demonstrate significant treatment benefit.
Nortriptyline for smoking cessation
A Cochrane systematic review, investigating the safety and efficacy of antidepressants for smoking cessation, yield the following results:
When used alone, nortriptyline produced a significant increase in long-term cessation based on six trials with 975 participants.
There was insufficient evidence that adding nortriptyline to nicotine replacement therapy provides an additional long-term effect.
Bupropion and nortriptyline appeared to be equally effective and similar efficacy to nicotine replacement therapy, although there was limited amount of data from direct comparisons.
While nortriptyline has the potential for serious side effects, none was demonstrated in the few small trials for smoking cessation.
Nortriptyline for patients with major depression and heart disease
There's a complex interactive relationship between depression and heart disease. Patients who became depressed after a heart attack had a fivefold increase in mortality during the first six months, according to a 1994 study. It has been thought that SSRIs are safer for patients with heart disease. In a study comparing Paroxetine and Nortriptyline for moderately depressed patients with ischemic heart disease, the two drugs seemed to have similar efficacy.
Nortriptyline has greater efficacy (92% vs 73%) in patients who completed treatment. However, more cardiovascular adverse events, including tachycardia, have been reported in nortriptyline and there were more cases of discontinuance compared to paroxetine. The researchers came to a conclusion that Paroxetine was as effective as nortriptyline but less likely to produce cardiovascular side effects.
Female Viagra: Safety and Effectiveness
November 1st 2018
In 2015, the FDA approved flibanserin (Addyi), also known as "female Viagra," for marketing after two previously failed attempts. Based on the theory that low libido in women, which is dubbed hypoactive sexual desire disorder (HSDD), is a result of an imbalance of neurotransmitters in the brain, flibanserin works by manipulating a variety of serotonin receptors, with some manipulated by excitation, while others are manipulated via inhibition. According to the prescribing information from Valeant Pharmaceuticals, the "mechanism of action in the treatment of premenopausal women with hypoactive sexual desire disorder is not known." How does flibanserin (Addyi) work? Yes, this is true. But this is true for a lot of medications. Almost one in five medications lack a well-defined mechanism of action. However, flibanserin does not work anything like sildenafil (Viagra), (vardenafil) Levitra, and tadalafil (Cialis) do, which is by causing increased bl ...
How Does Lupron Work During IVF
It has never been easy dealing with infertility, but thanks to fertility drugs, for example, Lupron and assisted reproductive technology like in vitro fertilization which has brought a ray of hope to many women struggling with conception challenges. For a clear understanding of how the drug works in IVF, it's advisable to comprehend the female reproductive system which is very complex; a number of hormones, organs, and glands must work as expected and at the right time. The hypothalamus gland located in your brain produces gonadotropin- releasing hormone (GnRH) on the first day of the monthly period. GnRH sends a signal to the pituitary gland to release follicle stimulating hormone (FSH) and luteinizing hormone (LH). As the pituitary gland is releasing both FSH and LH, it preserves most of the luteinizing hormone for later use. In the first days of the menses, follicle stimulating hormone is released the most. This is the start of what is termed as t ...
October 1st 2018
Thyroid Disease Different conditions of the thyroid gland mostly fall into these three categories: Hypothyroidism: This is the under-functioning of the thyroid. Hyperthyroidism: This is the over-functioning of the thyroid. Euthyroid: This refers to there being normal functioning of the thyroid. The thyroid The thyroid gland is part of the endocrine (gland) system in the body. The endocrine system is useful for doing things like making adrenalin, insulin, hormones, and many other important secretions. The thyroid gland makes thyroid hormones, and four pockets of parathyroid glands within it make parathormone (parathyroid hormone), which affects the balance of calcium for our bones. Removing the entire thyroid gland will also remove the parathyroid glands, a surgical caveat that became apparent as soon as thyroid surgery was developed. The thyroid itself is a butterfly-shaped gland that covers the trachea (windpipe) below the Adam's apple. Its main ...


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Are you sitting comfortably?
Off the record comments can be costly
Michelle steps up at Martin-Kaye
Employees need to be comfortable in the workplace but how far must their employers go to accommodate their needs?
Gemma Workman from Martin-Kaye Solicitors, in Euston Way, Telford, said employers should tread carefully when it comes to decisions about office furniture.
"Modern office chairs can be adjusted to accommodate a wide range of shapes and sizes, and they also come in different dimensions for both extra tall and smaller staff.
"But what if an employee would prefer to bring in their own seating because they found it more comfortable? In particular, how should an employer react if the choice isn't a traditional chair?
"If your staff work with a computer, their seating will be subject to health and safety regulations which set out certain requirements that standard chairs must have, including a height-adjustable seat and a backrest that can be adjusted for both height and tilt.
"Chairs must also be physically stable and allow for freedom of movement, but national guidance also says employers should consider the possibility of staff using unconventional chairs too.
"It says that employees suffering from back problems may benefit from a chair that comes with a fixed backrest or a seat that has no backrest at all, such as a kneeling chair. But whatever they choose, it must still be stable on the floor.
"This means employees will not be able to use exercise balls as seating for example, as although they can help strengthen core muscles, it's too easy for an employee to fall off and hurt themselves."
Miss Workman said before agreeing to anything, employers should find out exactly why their staff may want to bring in their own seating.
"Is their current chair uncomfortable because it's broken or because they are not a standard size? If so, then this is easily put right by ordering them a new chair. Or maybe the employee uses non-conventional seating at home and finds it more to their liking?
"If their request is purely down to personal choice, you should think carefully before giving them the go-ahead because it will make it very difficult to refuse any requests from other staff in the future."
Miss Workman said there may be some merit in the request if the employee has a pre-existing medical condition, or if they're recovering from a back or hip injury.
"But you should ask for a full assessment from an occupational health advisor before you say yes, and then providing their chosen chair is suitable, you can allow them to bring it in to the workplace."


Spanish photographer, graphic designer and CG artist sitted in Madrid, Spain. Founder of graphic design studio ILO DESIGNS focused in creating images by photography, illustration, CG art, 3d art and more.
Like the photos of Ignacio Leal Orozco?


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Seller does k225 offer returns. A brand-new, unused, unopened, undamaged item in its original packaging where packaging is applicable. We are not responsible for any accidents, delays or other issues that are the responsibility of the shipping service.
Get the item you ordered or get your money back. We are equipped with the most advanced tools used by the most experienced technicians. Customized transportation pallets – devices of k225 shapes, sizes and weght are transported on customized transportation pallets, making the delivery as safe as possible.
If the part you received is certify be defective,you can apply to get replace or get refund. Select a valid country. Any import fees or charges are paid by buyer. See all condition definitions – opens in a new window or tab Join other happy clients of RGB Elektronika!