text:
\section{Introduction}
Studying infinitesimal transformations of $J$-planar curves is a natural
question that has received increased attention during the last two decades.
Historically, the question can be traced back to classical mathematicians
like, for instance, Lie \cite{lie_1882,lie_1883}, Painlevé
\cite{painleve_1894}, Beltrami \cite{beltrami_1865} and Levi-Civita
\cite{levi-civita_1896} who investigated transformations of the differential
equations that describe trajectories of Hamiltonian systems up to
reparametrisations.

An infinitesimal symmetry of a system of ordinary differential equations
(ODEs) is a vector field on the space of dependent and independent variables
whose local flow sends solutions to solutions. It is well known that the
set of infinitesimal symmetries of a given ODE has the structure of a Lie
algebra. Studying the symmetry Lie algebras for ODEs is a powerful technique
by which the equivalence of two ODEs can be investigated, i.e.~whether one
can be mapped into the other by a change of variables. The dimension of the
symmetry Lie algebras of equivalent ODEs is the same.
For instance take the ODEs\footnote{We denote derivatives by subscripts.}
\begin{equation}\label{eqn:ode.2.dim}
	y_{xx}=-e^{-2x}y_x^3
\end{equation}
and
\begin{equation}\label{eqn:ode.3.dim}
	y_{xx}=\frac12 y_x-\frac12 e^{-2x}y_x^3\,,
\end{equation}
which are not equivalent because~\eqref{eqn:ode.2.dim} has a 2-dimensional
symmetry Lie algebra generated by
\begin{equation*}
	\partial_y\quad\text{and}\quad\partial_x+y\partial_y\,,
\end{equation*}
while \eqref{eqn:ode.3.dim} has a 3-dimensional symmetry Lie algebra
generated by
\begin{equation*}
	\partial_y\,,\quad \partial_x+y\partial_y
	\quad\text{and}\quad y\partial_x+\frac12 y^2\partial_y\,.
\end{equation*}
In other words, there is no change of variables $(x,y)\to
(x_\text{new}=x_\text{new}(x,y),y_\text{new}=y_\text{new}(x,y))$ sending
\eqref{eqn:ode.2.dim} into
\eqref{eqn:ode.3.dim}. Furthermore, neither \eqref{eqn:ode.2.dim} nor
\eqref{eqn:ode.3.dim} can be mapped, by a change of variables, to
\begin{equation}\label{eqn:ode.8.dim}
	y_{xx}=0
\end{equation}
as such an ODE admits an 8-dimensional Lie algebra of symmetries, whose
generators are
\begin{equation}\label{eqn:8.dim.Lie.alg}
	\partial_x\,,\quad\partial_y\,,\quad x\partial_x\,,\quad
	x\partial_y\,,\quad y\partial_x\,,\quad y\partial_y\,,\quad
	x^2\partial_x+xy\partial_y\quad \text{and}\quad
	xy\partial_x+y^2\partial_y\,.
\end{equation}
These examples are, of course, not chosen randomly, but are geometrically
motivated. Indeed, \eqref{eqn:ode.2.dim}, \eqref{eqn:ode.3.dim}
and~\eqref{eqn:ode.8.dim} are the equations of ``unparametrised geodesics''
for certain Riemannian metrics.
Specifically, \eqref{eqn:ode.8.dim} is the equation of \emph{lines} of
$\mathbb{R}^2$, i.e.~the geodesic curves of the Euclidean metric $dx^2+dy^2$.
Generators of the \emph{projective algebra} of $\mathbb{R}^2$ are given by
\eqref{eqn:8.dim.Lie.alg}: the local flow of any generator of
\eqref{eqn:8.dim.Lie.alg} sends lines of $\mathbb{R}^2$ into lines of
$\mathbb{R}^2$.
Likewise, Equation \eqref{eqn:ode.2.dim} is realised as the equation of
unparametrised geodesics for the metric $e^{4x}dx^2+e^{2x}dy^2$, Equation
\eqref{eqn:ode.3.dim} is realised as the equation of unparametrised
geodesic of $e^{3x}dx^2+e^xdy^2$. It follows, as a byproduct, that the
metrics $e^{4x}dx^2+e^{2x}dy^2$, $e^{3x}dx^2+e^{x}dy^2$ and $dx^2+dy^2$ are
pairwise non-isometric.

Let us return to~\eqref{eqn:ode.8.dim}.
By a suitable change of variables, \eqref{eqn:ode.8.dim} transforms into an ODE describing \emph{great circles} of the sphere. This insight is due to
Liouville~\cite{liouville_1889} and illustrated by Figure~\ref{fig:beltrami}.
In fact, up to changing variables, the ODE~\eqref{eqn:ode.8.dim} is the
equation of unparametrised geodesics of any 2-dimensional metric with
constant curvature.
\begin{figure}\label{fig:beltrami}
  \centering
	\begin{tikzpicture}

		\begin{scope}[xslant=0.5]
			\filldraw[fill=gray!10,opacity=0.2] (-3,1) -- (4,1) -- (4,-3) -- (-3,-3) -- cycle;
			\draw[thin] (-3,-2.5) -- (4,-2.5);
		\end{scope}

		\begin{scope}
			\clip(-3,-3) -- (-3,0) -- (-2,0) arc (180:360:2cm and 0.6cm) -- (3,0) -- (3,-3) -- (-3,-3);

				\filldraw[ball color=white] (0,0) circle [radius=2cm];
		\end{scope}

		\draw[black,thick] (2,0) arc [start angle=0,end angle=-180,x radius=2cm,y radius=17mm];

		\draw[solid] (2,0) arc [start angle=0,end angle=180,x radius=2cm,y radius=6mm];
		\draw (2,0) arc [start angle=0,end angle=-180,x radius=2cm,y radius=6mm];

		\shade[shading=radial, inner color=white, outer color=white!50!black, opacity=0.70] (2,0) arc (0:360:2cm and
		0.6cm);

		\draw[black,dashed] (0,0) -- (2,-2.5);
		\draw[black,dashed] (0,0) -- (1,-2.5);
		\draw[black,dashed] (0,0) -- (-1,-2.5);

		\draw[black,thick,-stealth] (1.1,-1.375) -- (2,-2.5);
		\draw[black,thick,-stealth] (0.6,-1.5) -- (1,-2.5);
		\draw[black,thick,-stealth] (-0.6,-1.5) -- (-1,-2.5);
	\end{tikzpicture}
\caption{The gnomonic (central) projection maps great circles to lines.}
\end{figure}
\medskip

In the present paper we ask a very similar question, yet in a different
context. Instead of geodesics and (pseudo-)Riemannian metrics, we consider
$J$-planar
curves and K\"ahler metrics (of arbitrary signature). The analog of
projective symmetry algebras, in
this context, are so-called c-projective algebras (see below for a proper
introduction). Our aim is to obtain, explicitly, the full,
non-trivial\footnote{By ``non-trivial'' we mean that there exists a
c-projective vector field that is essential, i.e.~not affine.} c-projective
algebras of K\"ahler surfaces.

\subsection{Preliminaries}

Let $(M,J,g)$ be a K\"ahler manifold of real dimension $2n$ and arbitrary
signature. This means that $M$ is a differentiable manifold with
(pseudo-)Riemannian metric $g$ and complex structure $J$, such that $\nabla
J=0$ where $\nabla$ is the Levi-Civita connection of $g$, and the 2-form
$\omega$ on $M$ defined by $\omega(X,Y)=g(J(X),Y)$ is a symplectic form. In
this context, the concept of unparametrised geodesic is replaced by that of
$J$-planar curve.
\begin{definition}
Let $(M,J,g)$ be a K\"ahler manifold. Let $\nabla$ be its Levi-Civita
connection. A curve $\gamma:I\subseteq\mathbb{R}\to \gamma(t)\in M$ such that
$$
	\nabla_{\dot\gamma}\dot\gamma\wedge\dot\gamma\wedge J\dot\gamma = 0
$$
is called a \emph{$J$-planar curve}.
\end{definition}
\noindent
It is easily verified that this definition is independent of parametrisation.
The following definition captures the corresponding infinitesimal symmetries.
\begin{definition}
Let $(M,J,g)$ be a K\"ahler manifold. A vector field is called \emph{c-projective} if its local flow maps
$J$-planar curves into $J$-planar curves.
\end{definition}
It is well known that the c-projective vector fields of a K\"ahler
manifold form a Lie algebra.
\emph{Affine} vector fields are simple examples of c-projective vector fields. Their local flow does not just preserve
$J$-planar curves but even the Levi-Civita connection $\nabla$.
C-projective vector fields that are not affine are called \emph{essential}.
We need two more definitions to state the main fact that serves as point of
departure for the present paper.
\begin{definition}
	Two K\"ahler metrics on the same manifold are \emph{c-projectively
	equivalent} if they	share the same $J$-planar curves.

	The collection of all K\"ahler surfaces that are c-projectively
	equivalent will be called a \emph{c-projective
	K\"ahler structure} $(M,J,[g])$.
\end{definition}
\noindent Indeed, $J$ is preserved by the flow of c-projective vector fields,
$\mathcal{L}_vJ=0$ if $v$ is a c-projective vector field.

We also recall the definition of holomorphic sectional curvature (HSC), which
is a restriction of the usual sectional curvature to $J$-invariant planes.
\begin{definition}
	Let $g$ be a K\"ahler metric with Riemann curvature $R$.
	The HSC of a $J$-invariant plane $\Pi=\langle v,Jv\rangle$ in $p\in M$ is
	defined by
	$$ K(\Pi) = \frac{R(v,Jv,v,Jv)}{||v||^4}\,, $$
	where $v\in T_pM$ is a vector at $p$.
	If the map $\Pi\mapsto K(\Pi)$ is constant and independent of $p$, then
	we say that $g$ has	constant HSC.
\end{definition}

\noindent In \cite{BMMR2015} a non-sharp local description is obtained for
c-projective K\"ahler structures admitting essential c-projective vector
fields.
\begin{fact}\label{fct:metrics}
	Let $(M,J,g_0)$ be a K\"ahler surface of non-constant HSC and with an
	essential c-projective vector field.
	Then, in a neighborhood of almost every point of $M$, there are local
	coordinates such that a certain metric $g\in[g_0]$ and its K\"ahler form $\omega$ defined by
	$\omega(X,Y)=g(J(X),Y)$ are as in (exactly) one of the cases below.
	\begin{enumerate}[label=(\roman*)]
		\item	\textbf{Liouville type:}
				In the new coordinates $(x_0,x_1,s_0,s_1)$,
				\begin{align*}
					g &= (\rho_0-\rho_1)(F_0^2dx_0^2+\veps F_1^2dx_1^2) \\
						&\quad +\frac1{\rho_0-\rho_1}\left[
								\left(\frac{\rho_0'}{F_0}\right)^2(ds_0+\rho_1ds_1)^2
								+\veps \left(\frac{\rho_1'}{F_1}\right)^2(ds_0+\rho_0ds_1)^2
							\right]
					\\
					\omega &= \rho_0'\,dx_0\wedge(ds_0+\rho_1\,ds_1)
								+\rho_1'\,dx_1\wedge(ds_0+\rho_0\,ds_1)
				\end{align*}
				where $\veps\in\{\pm1\}$ and where the univariate
				functions $\rho_0,\rho_1,F_0,F_1$ are given as in one of the
				cases below.
				\begin{enumerate}[label=(L\arabic*)]
					\item $\rho_i(x_i)=x_i$, $F_i=c_i\in\RR$
					\item $\rho_i(x_i)=c_ie^{(\beta-1)x_i}$, $F_i=d_ie^{-\frac12(\beta+2)x_i}$ with $c_i,d_i\in\RR$
							($\beta\ne1$)
					\item $\rho_i(x_i)=x_i$, $F_i=c_ie^{-\frac32x_i}$, $c_i\in\RR$
					\item $\rho_i(x_i)=-\tan(x_i)$, $F_i=\frac{c_ie^{-\frac32\beta x_i}}{\sqrt{|\cos(x_i)}}$,
							  $c_i\in\RR$
				\end{enumerate}
		\item	\textbf{Complex type:}
				In the new coordinates $(z,\bar z,s_0,s_1)$,
				\begin{align*}
					g &= \frac14\,(\bar\rho-\rho)(F^2dz^2-\bar F^2d\barz^2) \\
						&\quad +\frac4{\rho-\bar\rho}\left[
								\left(\frac{\rho'}{F}\right)^2(ds_0+\bar\rho ds_1)^2
								-\left(\frac{\bar\rho'}{\bar F}\right)^2(ds_0+\rho ds_1)^2
						\right]
					\\
					\omega &= \rho'\,dz\wedge(ds_0+\bar\rho\,ds_1)
						+\bar\rho'\,d\barz\wedge(ds_0+\rho\,ds_1)
				\end{align*}
				where the holomorphic functions $\rho,F$ are given as in one
				of the cases below (for brevity we denote
				$\rho'=\diff{\rho}{z}$ and $\rho'=\diff{\bar\rho}{\barz}$).
				\begin{enumerate}[label=(C\arabic*)]
					\item $\rho(z)=z$,
					$F=\varsigma=\varsigma_0+i\varsigma_1\in\CC$
					\item $\rho(z)=e^{(\beta-1)z}$,
					$F=\varsigma e^{-\frac12(\beta+2)z}$ with
					$\varsigma=\varsigma_0+i\varsigma_1\in\CC$
							($\beta\ne1$)
					\item $\rho(z)=z$, $F=\varsigma e^{-\frac32z}$,
					$\varsigma=\varsigma_0+i\varsigma_1\in\CC$
					\item $\rho(z)=-\tan(z)$, $F=\frac{\varsigma
					e^{-\frac32\beta z}}{\sqrt{\cos(z)}}$,
							$\varsigma=\varsigma_0+i\varsigma_1\in\CC$
				\end{enumerate}
		\item	\textbf{Degenerate type:}
				In the new coordinates $(x_0,x_1,s_0,s_1)$,
				\begin{align*}
					g &= -\rho\,h +\rho F^2 dx^2 +\frac1{\rho}\left(\frac{\rho'}{F}\right)^2\theta^2
					\\
					\omega &= -\rho\Omega +\rho' dx\wedge\theta
				\end{align*}
				where $h=\sum_{i,j}h_{ij}(s_0,s_1)ds_ids_j$ and
				$\theta=dy-\tau$, and
				where the functions $\rho,F,\tau$ are given as in one of the
				cases below.
				\begin{enumerate}[label=(D\arabic*)]
					\item $\rho(x_0)=\frac1{x_0}$,
					$F(x_0)=\frac{c_1}{\sqrt{|x_0|}}$
					and $\tau=s_0ds_1$. Moreover, $h=G(s_1)ds_0^2+\frac1{G(s_1)}ds_1^2$.
					\item[(D2a)] $\rho(x_0)=c_1e^{-3x_0}$, $F(x_0)=d_1$,
								$c_1,d_1\in\RR$, $\tau=s_0ds_1$, $h=G(s_1)ds_0^2+\frac1{G(s_1)}ds_1^2$.
					\item[(D2b)] $\rho(x)=c_1e^{(\beta-1)x_0}$,
					$F(x_0)=d_1e^{-\frac12(\beta+2)x_0}$
								($\beta\ne1$, $\beta\ne-2$),
								$c_1,d_1\in\RR$,\\
								$\tau=-\frac1{\beta+2}e^{-(\beta+2)s_0}G(s_1)ds_1$,
								$h=e^{-(\beta+2)s_0}G(s_1)(ds_0^2+ds_1^2)$
					\item[(D3)] $\rho(x_0)=\frac1{x_0}$,
					$F(x_0)=\frac{c_1e^{-\frac32x_0}}{\sqrt{|x_0|}}$,
								$\tau=-\frac13e^{-3s_0}G(s_1)ds_1$, $h=e^{-3s_0}G(s_1)(ds_0^2+ds_1^2)$
				\end{enumerate}
	\end{enumerate}
\end{fact}
\noindent
To avoid misconceptions, we would like to point out a few particulars of the
list in Fact~\ref{fct:metrics}:
\begin{itemize}
	\item
	Not all the metrics listed in Fact~\ref{fct:metrics} are of non-constant
	holomorphic sectional curvature. Theorem~\ref{thm:CHSC} below is
	addressing this point.
	\item
	Not every metric of non-constant holomorphic curvature with an essential
	c-projective vector field is isometric to one in the list, but every such
	metric can be expressed by using a metric $g$ from the list and a
	suitable companion metric $\hat g$ (detailed in
	Appendix~\ref{app:companion.metrics}) using the formula
	\begin{equation}\label{eqn:metric.construction}
		g[t_1,t_2] = \frac{\left(
						t_1\det(g)^{\tfrac16}g^{-1}
						+t_2\det(\hat g)^{\tfrac16}\hat g^{-1}\right)^{-1}
					}{
						\sqrt{\det\left(
							t_1\det(g)^{\tfrac16}g^{-1}
							+t_2\det(\hat g)^{\tfrac16}\hat g^{-1}
						\right)}
					} \,,\quad t_i\in\mathbb{R}\,.
	\end{equation}
	The problem of describing these individual metrics, up to isometries, is
	not solved in the current paper, but is going to be addressed in a
	separate paper by the authors \cite{MSV2022}.
	For our purposes it is sufficient to state the following: The three types
	are disjoint, so a metric satisfying the hypothesis of
	Fact~\ref{fct:metrics} is either of Liouville, complex or degenerate
	type.

	\item
	Metrics obtained using Formula \eqref{eqn:metric.construction} have the same c-projective algebra as
	the metric~$g$ itself. In particular, for metrics of constant holomorphic
	sectional curvature, the algebra of c-projective vector fields is
	well understood and always isomorphic to that of the Fubini-Study metric.
	For the metrics of non-constant HSC which admit a non-affine c-projective
	vector field, we determine the full c-projective algebras in this paper.

	\item
	Fact~\ref{fct:metrics} provides a characterisation modulo c-projective
	equivalence. As a consequence, there are metrics in the list of
	Fact~\ref{fct:metrics} that admit only c-projective vector fields that
	are affine. However, generically, the metrics c-projectively equivalent
	to the ones listed in Fact~\ref{fct:metrics} admit essential c-projective
	vector fields.
	Note
	that c-projective algebras are invariant under c-projective
	transformations.
\end{itemize}
\noindent Before turning to our main results, let us consider the case of
constant HSC more closely. Such metrics are c-projectively
equivalent to the Fubini-Study metric \cite{FKMR2010,kruglikov_2016}.
Their c-projective algebras are therefore isomorphic to that of the
Fubini-Study metric.
\begin{example}[Fubini-Study metric on $\CC\mathbb{P}^2$]
\label{ex:fubini-study}
Let $z_1=x+iy$ and $z_2=s+it$ be complex coordinates on $\CC\mathbb{P}^2$.
The Fubini-Study metric on $\CC\mathbb{P}^2$ is locally described by
\begin{equation}\label{eqn:Fubini.metric.local} g=\frac{(1+\sum_iz_i\bar{z}_i)\sum_jdz_jd\bar{z}_j-\sum_{i,j}\bar{z}_jz_idz_jd\bar{z}_i}{(1+\sum_iz_i\bar{z}_i)^2}\,.
\end{equation}
The complex structure $J$ in these coordinates takes the standard form
\begin{equation*}
	J=
	\left(
	\begin{array}{cccc}
		0 & 1 & 0 & 0
		\\
		-1 & 0 & 0 & 0
		\\
		0 & 0 & 0 & 1
		\\
		0 & 0 & -1 & 0
	\end{array}
	\right)
\end{equation*}
The c-projective algebra of~\eqref{eqn:Fubini.metric.local} is isomorphic
to~$\mathfrak{sl}(3,\CC)$ \cite{MS1983,FKMR2010}.
An explicit realisation of this algebra in terms of 16 generating vector
fields can be found in Appendix~\ref{app:fubini-study}.
\end{example}

\noindent
With this example in mind, the purpose of the present paper has to be
twofold: We first identify the metrics in Fact~\ref{fct:metrics} that have
constant holomorphic sectional curvature, and then we determine the full
c-projective algebras of those metrics in Fact~\ref{fct:metrics} that have
non-constant holomorphic sectional curvature.
In particular, we obtain the following four main theorems.
\begin{theorem}\label{thm:CHSC}
	Among the metrics in Fact~\ref{fct:metrics} the following have constant
	HSC:
	\begin{enumerate}
		\item[(L1)] Metrics of L1 type have constant HSC if $\varepsilon=-1$
		and $c_1=\pm c_0$.
		\item[(L2)] Metrics of L2 type have constant HSC if $\beta=-\tfrac12$
		and $c_1=\varepsilon c_0\frac{d_1^2}{d_0^2}$, or if $\beta=-2$,
		$\varepsilon=-1$ and $d_1=\pm d_0$.
		\item[(C1)] Metrics of C1 type have constant HSC if either
		$\Re(\varsigma)=0$
		or $\Im(\varsigma)=0$.
		\item[(C2)] Metrics of C2 type have constant HSC if $\beta=-\tfrac12$
		and either $\Re(\varsigma)=0$ or $\Im(\varsigma)=0$, or if $\beta=-2$
		and either
		$\Re(\varsigma)=0$ or $\Im(\varsigma)=0$.
		\item[(D1)] Metrics of D1 type have constant HSC if the metric $h$
		has
		vanishing Gau{\ss} curvature.
		\item[(D2a)] Metrics of D2a type have constant HSC if the metric $h$
		has
		Gau{\ss} curvature exactly $\frac{9}{d_0^2}$.
	\end{enumerate}
	All other metrics in Fact~\ref{fct:metrics} are of non-constant HSC.
\end{theorem}
\noindent
Having answered the question which metrics of Fact~\ref{fct:metrics} have
constant HSC, we now turn to determining the c-projective vector fields of
those metrics with non-constant HSC.
We begin with Liouville type metrics.
\begin{theorem}\label{thm:Liouville.main}
	Let $g$ be a K\"ahler surface covered by Fact~\ref{fct:metrics}. Assume
	it is of non-constant HSC and that it is of Liouville type.
	Then its c-projective algebra is isomorphic to one of the following.
	\begin{enumerate}[label=(L\arabic*)]
		\item
		The c-projective algebra is 4-dimensional,
		\begin{align*}
			\projalg(g) = \big\langle\quad
				\del_{x_0}+\del_{x_1}-s_1\del_{s_0}\,,\quad
				x_0\del_{x_0}+x_1\del_{x_1}+2s_0\del_{s_0}+s_1\del_{s_1}\,,\quad
				\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
		\end{align*}
		\item
		The algebra is either 3-dimensional or 4-dimensional.
		\begin{enumerate}[label=(\roman*)]
			\item If $\beta\ne0$ and $c_1^2d_0^2+\varepsilon c_0^2d_1^2\ne0$,
			then the c-projective algebra is 3-dimensional,
			\begin{align*}
				\projalg(g) = \big\langle\quad
					\del_{x_0}+\del_{x_1}-(\beta+2)s_0\del_{s_0}
							-(2\beta+1)s_1\del_{s_1}\,,\quad
					\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
			\end{align*}

			\item If $\beta=0$, or $c_1^2d_0^2+\varepsilon c_0^2d_1^2=0$ with
			$\varepsilon=-1$, then the c-projective algebra is 4-dimensional,
			\begin{align*}
				\projalg(g) = \big\langle\quad
						\del_{x_0}+\del_{x_1}
							-2s_0\del_{s_0}
							-s_1\del_{s_1}\,,\quad
						\frac1{c_0}e^{x_0}\del_{x_0}
							+\frac1{c_1}e^{x_1}\del_{x_1}
							+s_1\del_{s_0}\,,\quad
						\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
			\end{align*}
		\end{enumerate}
		\item
		The c-projective algebra is 3-dimensional,
		\begin{align*}
			\projalg(g) = \big\langle\quad
					\partial_{x_0}+\partial_{x_1}
							-(3s_0+s_1)\partial_{s_0}
							-3s_1\partial_{s_1}\,,\quad
					\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
		\end{align*}
		\item
		The c-projective algebra is 3-dimensional,
		\begin{align*}
			\projalg(g) = \big\langle\quad
					\partial_{x_0}+\partial_{x_1}
							-(3\beta s_0-s_1)\partial_{s_0}
							-(s_0+3\beta s_1)\partial_{s_1}\,,\quad
					\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
		\end{align*}
	\end{enumerate}
\end{theorem}
\noindent
Next, we consider complex type metrics.
\begin{theorem}\label{thm:complex.main}
	Let $g$ be a K\"ahler surface covered by Fact~\ref{fct:metrics}. Assume
	it is of non-constant HSC and that it is of complex type.
	Then its c-projective algebra is isomorphic to one of the following.
	\begin{enumerate}[label=(C\arabic*)]
		\item
		The c-projective algebra is 4-dimensional:
		\begin{align*}
			\projalg(g) = \big\langle\quad
					\del_z+\del_{\barz}-s_1\del_{s_0}\,,\quad
					z\del_z+\barz\del_{\barz}
							+2s_0\del_{s_0}+s_1\del_{s_1}\,,\quad
					\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
		\end{align*}
		\item
		The c-projective algebra is either of 3-dimensional or 4-dimensional,
		\begin{enumerate}[label=(\roman*)]
			\item If $\beta\ne0$ and $\varsigma^2\ne\bar\varsigma^2$,
			i.e.~neither $\varsigma_0=0$ nor $\varsigma_1=0$, then the
			c-projective algebra is 3-dimensional,
			\begin{align*}
				\projalg(g) = \big\langle\quad
						\del_{z}+\del_{\barz}
							-(\beta+2)s_0\del_{s_0}
							-(2\beta+1)s_1\del_{s_1}\,,\quad
						\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
			\end{align*}

			\item If $\beta=0$ or $\varsigma^2=\bar\varsigma^2$ (i.e.~either
			$\varsigma_0=0$ or $\varsigma_1=0$), then the c-projective
			algebra is 4-dimensional,
			\begin{align*}
				\projalg(g) = \big\langle\quad
					\del_z+\del_{\barz}-2s_0\del_{s_0}-s_1\del_{s_1}\,,\quad
					e^{z}\del_{z}+e^{\barz}\del_{\barz}+s_1\del_{s_0}\,,\quad
					\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
			\end{align*}
		\end{enumerate}
		\item
		The c-projective algebra is 3-dimensional,
		\begin{align*}
			\projalg(g) = \big\langle\quad
					\partial_{z}+\partial_{\barz}-(3s_0+s_1)\partial_{s_0}
							-3s_1\partial_{s_1}\,,\quad
					\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
		\end{align*}
		\item
		The c-projective algebra is 3-dimensional,
		\begin{align*}
			\projalg(g) = \big\langle\quad
					\partial_z+\partial_{\barz}-(3s_0+s_1)\partial_{s_0}
								-3s_1\partial_{s_1}\,,\quad
					\del_{s_0}\,,\quad\del_{s_1}\quad\big\rangle\,.
		\end{align*}
	\end{enumerate}
\end{theorem}
\noindent
Lastly, there are the metrics of degenerate type.
\begin{theorem}\label{thm:degenerate.main}
	Let $g$ be a K\"ahler surface covered by Fact~\ref{fct:metrics}. Assume
	it is of non-constant HSC and that it is of Degenerate type.
	Then its c-projective algebra is isomorphic to one of the following.
	\begin{enumerate}[label=(D\arabic*)]
		\item
		The c-projective algebra is of dimension 3, 4 or 5.
		\begin{enumerate}[label=(\roman*)]
			\item If $h$ has constant (non-zero) Gau{\ss} curvature,
			$$ h = G(s_1)ds_0^2+\frac{ds_1^2}{G(s_1)}\,,\qquad
			G(s_1) = \kappa s_1^2+\mu_1s_1+\mu_2\,,\quad
			\kappa\ne0\,, $$
			then the c-projective algebra is 5-dimensional (see
			Section~\ref{sec:D1.results}).

			\item If $h$ admits a 2-dimensional homothetic symmetry algebra,
			 then
			 $$ h = G(s_1)ds_0^2+\frac{ds_1^2}{G(s_1)}\,,\qquad
			 	G(s_1) = \kappa(\mu_1s_1+\mu_2)^{\frac{2(\mu_1+1)}{\mu_1}} $$
			 with $\projalg(h)=\langle\del_{s_0}\,,~~
			 		(\mu_1+2)s_0\del_{s_0}-(\mu_1s_1+\mu_2)\del_{s_1}\rangle$.
			 Then the c-projective algebra is 4-dimensional,
			 \begin{align*}
			 	\projalg(g) &= \big\langle\quad
			 			\partial_{x_0}\,,\quad
			 			\partial_{x_1}\,,\quad
			 			s_1\partial_{x_1}+\partial_{s_0}\,,
			 			\\
			 			&\qquad\quad
				 		2x_0\partial_{x_0}+2x_1\partial_{x_1}
				 		+(\mu_1+2)s_0\partial_{s_0}
			 			-(\mu_1s_1+\mu_2)\partial_{s_1}\quad
				 	\big\rangle\,.
			 \end{align*}

			\item If neither of the previous two cases is assumed, then the
			c-projective algebra is 3-dimensional,
			\begin{align*}
				\projalg(g) = \big\langle\quad
					\del_{x_0}\,,\quad\del_{x_1}\,,\quad
					s_1\del_{x_1}+\del_{s_0}\quad\big\rangle\,.
			\end{align*}
		\end{enumerate}
		\item[(D2a)]
		The c-projective algebra is of dimension 3 or 5.
		\begin{enumerate}[label=(\roman*)]
			\item If the metric $h$ has constant Gau{\ss} curvature,
			then the c-projective algebra of $g$ is
			5-dimensional (see Section~\ref{sec:D2a.results}).

			\item If $h$ admits a 2-dimensional or 1-dimensional homothetic
			symmetry algebra, then the c-projective algebra is 3-dimensional,
			\begin{align*}
				\projalg(g) = \big\langle\quad
				\del_{x_0}\,,\quad\del_{x_1}\,,\quad
				s_1\del_{x_1}+\del_{s_0}\quad\big\rangle\,.
			\end{align*}

		\end{enumerate}
		\item[(D2b)]
		The c-projective algebra is of dimension 2, 3 or 4.
		\begin{enumerate}[label=(\roman*)]
			\item If $h$ has constant Gau{\ss} curvature, then it is already
			flat. The c-projective algebra of $g$ thus is 4-dimensional (see
			Section~\ref{sec:D2b.results}).

			\item If $h$ admits a 2-dimensional homothetic
			symmetry algebra, then the c-projective algebra is 3-dimensional (see
			Section~\ref{sec:D2b.results}).

			\item If $h$ has 1-dimensional homothetic algebra, then the
			c-projective algebra of $g$ is 2-dimensional,
			$$ \projalg(g) = \big\langle\quad
					\del_{x_0}+(\beta+2)x_1\del_{x_1}+\del_{s_0}\,,\quad
					\del_{x_1}\quad\big\rangle\,. $$
		\end{enumerate}
		\item[(D3)]
		The c-projective algebra is of dimension 2, 3 or 4.
		\begin{enumerate}[label=(\roman*)]
			\item If $h$ has constant Gau{\ss} curvature, then $h$ is already
			flat. The c-projective algebra of $g$ is thus
			4-dimensional (see Section~\ref{sec:D3.results}).

			\item If $h$ admits a 2-dimensional homothetic
			symmetry algebra, then the c-projective algebra is 3-dimensional (see
			Section~\ref{sec:D3.results}).

			\item If $h$ has a 1-dimensional homothetic algebra, then the
			c-projective algebra of $g$ is 2-dimensional,
			$$ \projalg(g) = \big\langle\quad
				\del_{x_0}+3x_1\del_{x_1}+\del_{s_0}\,,\quad
				\del_{x_1}\quad\big\rangle\,. $$
		\end{enumerate}
	\end{enumerate}
\end{theorem}

\noindent In the remainder of the paper we shall prove
Theorems~\ref{thm:CHSC}, \ref{thm:Liouville.main}, \ref{thm:complex.main}
and~\ref{thm:degenerate.main}.
The sections are organised as follows:
we begin by proving Theorem~\ref{thm:CHSC} in Section~\ref{sec:CHSC}.
Section~\ref{sec:method} outlines the methodology we apply for the proofs of
Theorems~\ref{thm:Liouville.main}, \ref{thm:complex.main}
and~\ref{thm:degenerate.main}. The proofs are then completed in
Sections~\ref{sec:Liouville},~\ref{sec:complex} and~\ref{sec:degenerate},
respectively.
The paper is complemented by an appendix, where we summarise some background
material that will likely help readers not yet familiar with the literature,
particularly~\cite{BMMR2015}. However, the appendix is not needed to
follow the main text of the paper.

\section{Constant holomorphic sectional curvature}\label{sec:CHSC}

\noindent
Theorem~\ref{thm:CHSC} identifies those Kähler metrics in
Fact~\ref{fct:metrics} that are of constant HSC. In the current section we
prove this theorem.
It should be noted that K\"ahler manifolds of constant HSC are well
understood, e.g.~\cite{FKMR2010,BMMR2015,kruglikov_2016,CEMN2020}.
\begin{fact}\label{fct:CHSC}
	Let $g$ be a K\"ahler surface of constant HSC. Then the c-projective
	algebra of $g$ is (locally) isomorphic to that of the Fubini-Study metric
	outlined in Example~\ref{ex:fubini-study}.
\end{fact}
\noindent
For higher dimensions, a similar statement holds true.
An important property in the theory of c-projective K\"ahler structures is
the degree of mobility. Without going into detail, the degree of mobility is
the minimal number of linearly independent K\"ahler metrics needed to express
any metric $\hat g$ c-projectively equivalent to a given metric $g$ from the
structure.
Let $[g]$ be a c-projective K\"ahler structure. Then it can be shown that
there is a minimal number $N$ such that $[g]$ is parametrised by $N$ linearly
independent K\"ahler metrics $g_k$ ($1\leq k\leq N$) via
\[
	[g] = \left\{\quad
		\frac{
			(\sum t_k\det(g_k)^{\tfrac16}g_k^{-1})^{-1}
		}{
			\sqrt{\det(\sum t_k\det(g_k)^{\tfrac16}g_k^{-1})}
		}\quad : \quad
		t_k\in\RR
		\quad\right\}
\]
This number $D([g])=N$ is called the degree of mobility of $[g]$. We also say
that $g\in[g]$ has degree of mobility $D(g)=N$.
\begin{lemma}[\cite{MD1978,MS1983,BMMR2015}]\label{la:mobility}~

	\noindent(i) Let $g$ be a K\"ahler metric. It is of constant HSC if and
	only if $D(g)=(n+1)^2$.

	\noindent(ii) Let $g$ be a K\"ahler surface of non-constant HSC, which
	admits an essential c-projective vector field. Then $D(g)=2$.
\end{lemma}
\begin{proof}
	Claim~(i) is proven in~\cite{MD1978,MS1983} and claim~(ii)
	in~\cite{BMMR2015}. See also~\cite{MR2015}.
\end{proof}

For what follows here and in the following sections, we shall also need a
certain (1,1)-tensor field.
In order to introduce it, let $g$ and $\hat g$ be c-projectively equivalent
metrics (not necessarily of dimension $2n=4$, but in the following we will
only consider K\"ahler surfaces). We define
\begin{equation}\label{eqn:L}
	L[g,\hat g] = \left|\frac{\det(g)}{\det(\hat g)}\right|^{\frac1{2(n+1)}}\,
	\hat g^{-1}g\,,
\end{equation}
see also Appendix~\ref{app:companion.metrics}.
We proceed in two steps:
\begin{enumerate}
	\item
		In the light of Lemma~\ref{la:mobility}, in the present section we
		are looking for metrics listed in Fact~\ref{fct:metrics} whose degree
		of mobility is larger than two.
		A necessary criterion is obtained as follows: If the degree of
		mobility of any of the metrics in Fact~\ref{fct:metrics} is larger
		than two, then $L$ satisfies the (simplified) Sinjukov
		equations\footnote{We remark that the converse is not true: not every
		$V_n(B)$ geometry has degree of mobility larger than two.} of a
		so-called $V_4(B)$ geometry \cite{Mikes1998},
		\begin{subequations}\label{eqn:sinjukov}
		\begin{align}
			\label{eqn:sinjukov.1}
			\nabla_k L_{ij} &= g_{ik}\Lambda_{j} + g_{jk}\Lambda_{i}
										+\omega_{ik}\Lambda_{a}J\indices{^a_j}
										+\omega_{jk}\Lambda_{a}J\indices{^a_i}
			\\
			\label{eqn:sinjukov.2}
			\nabla_{j}\Lambda_{i} &= \mu g_{ij}+BL_{ij}
			\\
			\label{eqn:sinjukov.3}
			\nabla_k\mu &= 2B\Lambda_{k}
		\end{align}
		\end{subequations}
		where $\Lambda_{k}=\nabla_k\Lambda$, $\Lambda=\frac14\tr(L)$.
		We first determine necessary conditions for the parameters
		$\beta,\varsigma,c_i,d_i$. Indeed, if $g$ has constant HSC, it needs
		to satisfy
		this system of partial differential equations (PDEs), where $L$ is
		computed from $g$ and a suitable companion metric as in
		Appendix~\ref{app:companion.metrics}.
	\item
		For the special parameter configurations identified in the first
		step, we can check the HSC of the metrics directly, deciding if $g$
		has constant HSC and thus completing the proof.
\end{enumerate}
Since in the first step we merely need a necessary criterion, and since the
different types of metrics in Fact~\ref{fct:metrics} are qualitatively
different, we split the discussion according to the type of the metrics and
discuss each case separately.

\subsection{Cases L1--L4}
We proceed in two steps.
First, we obtain a necessary condition from Sinjukov's special PDE system.
Note that the (1,1)-tensor $L$ in these cases is diagonal in the coordinates
of Fact~\ref{fct:metrics}.
However, we note that $\nabla\Lambda$ generically is not diagonal, as its
$(2,3)$-entry does not automatically vanish (along with its $(3,2)$-entry
for symmetry reasons). Thus we obtain the condition that this component
vanishes in the case of $D(g)\geq3$.

Second, we check by a direct computation if, with this necessary condition
fulfilled, the metric has indeed constant HSC or not.
The results are detailed in the following table.\medskip

\begin{center}
\begin{tabulary}{\linewidth}{l|p{5cm}|p{7cm}}
	Metric & Sinjukov condition & constant HSC? \\
	\toprule\toprule
	L1 & $c_0^2\varepsilon+c_1^2=0$ & Yes, if $\varepsilon=-1$ and $c_1=\pm c_0$ \\
	\midrule
	L2 & $\beta=-\frac12$, $c_1d_0^2\varepsilon+c_0d_1^2=0$
		& Yes, if $\varepsilon=\pm1$ and $c_1=\varepsilon c_0\frac{d_1^2}{d_0^2}$ \\
	\cmidrule{2-3}
		& $\beta=-2$, $d_0^2\epsilon+d_1^2=0$
		& Yes, if $\varepsilon=-1$ and $d_1=\pm d_0$ \\
	\midrule
	L3 & Never possible & No \\
	\midrule
	L4 & $c_0^2\varepsilon+c_1^2=0$ & No
\end{tabulary}
\end{center}

\subsection{Cases C1--C4}
The reasoning is identical to that in the cases C1--C4, with the formal
replacements $\varepsilon\to-1$, and $c_0\to
\varsigma=\varsigma_0+i\varsigma_1$, $c_1\to\bar\varsigma$ in the case (C1
and C4) and $\varepsilon=-1$ and
$c_0\to1$, $c_1\to1$ and $d_0\to\varsigma$ , $d_1\to\bar\varsigma$ (C2 and
C3).
We arrive at the following table:\medskip

\begin{center}
\begin{tabulary}{\linewidth}{l|p{6cm}|p{8cm}}
	Metric & Sinjukov condition & constant HSC? \\
	\toprule\toprule
	C1 & $\varsigma^2=\bar\varsigma^2=0$ (where
	$\varsigma=\varsigma_0+i\varsigma_1$) & Yes, if $\varsigma_0=0$ or
	$\varsigma_1=0$ \\
	\midrule
	C2 & $\beta=-\frac12$, $\varsigma_0\varsigma_1=0$
	($\varsigma=\varsigma_0+i\varsigma_1$)
		& Yes, if $\beta=-\frac12$ and either $\varsigma_0=0$ or
		$\varsigma_1=0$ \\
	\cmidrule{2-3}
		& $\beta=-2$, $\varsigma_0\varsigma_1=0$
		($\varsigma=\varsigma_0+i\varsigma_1$)
		& Yes, if $\beta=-2$ and either $\varsigma_0=0$ or $\varsigma_1=0$ \\
	\midrule
	C3 & Never possible & No \\
	\midrule
	C4 & No constraint & No
\end{tabulary}
\end{center}

\subsection{Cases D1--D3}
Consider~\eqref{eqn:sinjukov.2} locally as a matrix equation.
It is diagonal except for the $(1,3)$-component, from which we obtain the
equation
$$
	BF^3\rho^3+2\rho F'\rho'+2F\rho'^2-2F\rho\rho'' = 0
$$
for some constant $B\in\RR$.
Solving for $B$ and taking one derivative w.r.t.~$x_0$, we obtain an
explicit condition depending on $\rho$, $F$ and their derivatives only:
\[
3\rho^2\,F'^2\rho' - F\rho^2\,F''\rho'
+ 4F\rho\,F'\rho'^2 + 3F^2\rho'^3
- 3F\rho^2\,F'\rho'' - 4F^2\rho\,\rho'\rho''
+ F^2\rho^2\,\rho''' = 0\,.
\]
Evaluating this condition for each of the cases D1--D3, and then examining
for constant HSC, we arrive the following table.\medskip

\begin{center}
\begin{tabulary}{\linewidth}{l|p{6cm}|p{6cm}}
	Metric & Sinjukov condition & constant HSC? \\
	\toprule\toprule
	D1 & The constraint for higher mobility is trivial, i.e.~no constraint
		& Yes, if $h$ has vanishing Gau{\ss} curvature \\
	\midrule
	D2a & The constraint for higher mobility is trivial, i.e.~no constraint
		& Yes, if $h$ has Gau{\ss} curvature $-\frac9{d_0^2}$ \\
	\midrule
	D2b & Would require $\beta=-2$ or $\beta=1$, which are forbidden
		& No \\
	\midrule
	D3 & Never possible & No
\end{tabulary}
\end{center}

We have therefore proven Theorem~\ref{thm:CHSC}.

\section{Method for Theorems~\ref{thm:Liouville.main}, \ref{thm:complex.main}
and \ref{thm:degenerate.main}}\label{sec:method}

We now turn to the main purpose of this paper and compute the c-projective
algebras of the metrics of non-constant HSC in Fact~\ref{fct:metrics}.
Methodologically, we use an approach similar to that employed in
\cite{BMMR2015}, \cite{solodovnikov_1956} and \cite{MV2022}.
Lemma~\ref{la:mobility} ensures that the degree of mobility is precisely two
if a metric from Fact~\ref{fct:metrics} has non-constant HSC.
Metrics with constant HSC have, of course, already been identified in
Theorem~\ref{thm:CHSC}.
The method will subsequently be adapted for the individual cases of Liouville,
complex and degenerate type in the following three sections.
The following lemma is useful, see e.g.~\cite{bolsinov_2011,BMR2021,MV2022}.
\begin{lemma}
	Let $g$ and $\hat g$ be c-projectively equivalent K\"ahler surfaces which
	are
	not of constant HSC.
	Let $v$ be a c-projective vector field of $g$.
	Then there are constants $a_{ij}\in\RR$ ($1\leq i,j\leq 2$) such that
	\begin{align}
		\lie_vL &= -a_{01}L^2+(a_{11}-a_{00})L+a_{10}\Id
		\label{eqn:LvL}
		\\
		\lie_vg &= a_{00}(n+1)\,\Id+a_{01}\,(L+\tr(L)\Id)\,.
		\label{eqn:Lvg}
	\end{align}
	Particularly, if $v$ is essential, then $a_{01}\ne0$, and if $v$ is
	properly homothetic for $g$, then $a_{01}=0$ and $a_{00}\ne0$. If $v$ is
	Killing for $g$, then $a_{00}=a_{01}=0$.
\end{lemma}

We prove the main theorems by solving the equations~\eqref{eqn:LvL}
and~\eqref{eqn:Lvg}.
A partial solution of~\eqref{eqn:LvL} is well known and can be exploited in
the following. It concerns eigenvalues of $L$ and is partly folklore. An
explicit proof can be found in~\cite{MV2022}.
\begin{lemma}\label{la:vf}
	Let $g$ and $\hat g$ be c-projectively equivalent K\"ahler surfaces which
	are not of constant HSC, and let $v$ be a
	c-projective vector field.
	Any eigenvalue $f$ of~\eqref{eqn:L} satisfies
	$$
		v(f) = -a_{01}f^2+(a_{11}-a_{00})f+a_{10}\,.
	$$
\end{lemma}
Our study below is going to be local, and therefore the
equations~\eqref{eqn:LvL} and~\eqref{eqn:Lvg} can be represented by $4\by4$
matrices. In fact, for the metrics of Fact~\ref{fct:metrics} these matrices
have a block structure consisting of four $2\by2$ matrices each.
\begin{align*}
	\text{Eq.~\eqref{eqn:LvL}:}\qquad &&
	\lie_vL+a_{01}L-(a_{11}-a_{00})L-a_{10}\Id&=0
	\qquad\leftrightarrow&
	\begin{pmatrix} (*a) & (*b) \\ (*c) & (*d) \end{pmatrix}=0
	\\
	\text{Eq.~\eqref{eqn:Lvg}:}\qquad &&
	\lie_vg-a_{00}(n+1)\,\Id-a_{01}\,(L+\tr(L)\Id)&=0
	\qquad\leftrightarrow&
	\begin{pmatrix} (*A) & (*B) \\ (*C) & (*D) \end{pmatrix}=0
\end{align*}

\section{Proof of Theorem~\ref{thm:Liouville.main} (Liouville type)}
\label{sec:Liouville}

We proceed in two distinct steps. First we prove the following auxiliary
result, which allows us to reduce the problem to the univariate realm.
\begin{lemma}\label{la:Liouville}
	Any projective symmetry of a Liouville metric in the coordinates of
	Fact~\ref{fct:metrics} is of the form
	\[
		v = v^0(x_0)\partial_{x_0}+v^1(x_1)\partial_{x_1}
			+v^2(s_0,s_1)\partial_{s_0}+v^3(s_0,s_1)\partial_{s_1}\,.
	\]
	In particular, if $F_1^2\rho_0'^2+\varepsilon F_0^2\rho_1'^2\ne0$, then
	even
	\begin{align*}
		v^2(s_0,s_1) &= \mu s_0-a_{10}s_1+\nu
		\\
		v^3(s_0,s_1) &= -a_{01}s_0+(\mu+a_{00}-a_{11})s_1+\nu_0
	\end{align*}
	for constants $\mu,\nu$ and $\nu_0$.
\end{lemma}
Using this lemma, the proof of Proposition~\ref{prop:degenerate} is then
going to continue along the following lines:
The remaining algebraic conditions are of the form
\begin{equation}\label{eqn:liouville.cases.algebraic.condition}
	\sum_\mu\sum_\nu\ c_{\mu\nu} F_\mu(x_0)\hat F_\nu(x_1) = 0
\end{equation}
where the $c_{\mu\nu}$ depend on the parameters $a_{ij}$ and on the
parameters of the metric.
Without losing generality, we may take the functions $F_\mu$ (respectively
$\hat F_\nu$) to be linearly independent functions (otherwise use linear
dependence to replace some functions until linear independence is
achieved, changing the functions $F_\mu$ and $\hat F_\nu$
in~\eqref{eqn:liouville.cases.algebraic.condition}).
We hence conclude
\[
	c_{\mu\nu} = 0 \quad\forall\ \mu,\nu\,,
\]
and then solve all remaining conditions, arriving eventually at
Theorem~\ref{thm:Liouville.main}.

\subsection{Proof of the lemma}
We begin proving Lemma~\ref{la:Liouville}.
For the metric $g$ we have, in the coordinates $(x_0,x_1,s_0,s_1)$,
$$
	L = \begin{pmatrix}
			\rho_0(x_0) &&& \\
			& \rho_1(x_1) && \\
			&& \rho_0+\rho_1 & \rho_0\rho_1 \\
			&& -1 & 0
		\end{pmatrix}
$$
and take the c-projective vector field as
$
	v = (v^0,v^1,v^2,v^3)
$
where each component is a function $v^i=v^i(x_0,x_1,s_0,s_1)$ in four
variables.
We first take the (0,0) and (1,1) components of~$(*a)$, i.e.\ which are the
equations from Lemma~\ref{la:vf}.
Since $\rho_0$ and $\rho_1$ are non-constant, we find ($j\in\{0,1\}$)
\begin{align*}
	v^j &= -\frac{
				a_{01}\rho_j(x_j)^2 + (a_{00}-a_{11})\rho_j(x_j) - a_{10}
			}{
				\frac{\partial\rho_j}{\partial x_j}
			}
\end{align*}
and conclude $v^0=v^0(x_0)$ as well as $v^1=v^1(x_1)$.
Resubstituting into~\eqref{eqn:LvL}, the block $(*b)$ is redundant, and the
block $(*c)$ yields
\begin{align*}
	\rho_0(x_0)\frac{\partial v^3}{\partial x_0}
	+\frac{\partial v^2}{\partial x_0} = 0
	\\
	\rho_1(x_1)\frac{\partial v^3}{\partial x_1}
	+\frac{\partial v^2}{\partial x_1} = 0
\end{align*}
Moreover, then the block $(*B)$ implies
\begin{align*}
	\frac{\partial\rho_0(x_0)}{\partial x_0}
	\frac{\partial v^3}{\partial x_0} = 0
	\\
	\frac{\partial\rho_1(x_1)}{\partial x_1}
	\frac{\partial v^3}{\partial x_1} = 0
	\\
	(\rho_0-\rho_1)\,
	\frac{\partial\rho_1}{\partial x_1}
	\frac{\partial v^3}{\partial x_1} = 0
\end{align*}
We conclude $v^3=v^3(s_0,s_1)$ and then $v^2=v^2(s_0,s_1)$.

Now revisit the block $(*d)$. From the (3,3) and (3,2) components, we obtain
\begin{align*}
	\left(a_{01}+\frac{\partial v^3}{\partial s_0}\right)\rho_0\rho_1
	+\left(a_{10}+\frac{\partial v^2}{\partial s_1}\right) &= 0
	\\
	\left(b+\frac{\partial v^3}{\partial s_0}\right)(\rho_0+\rho_1)
	+\left(a_{00}-a_{11}+\frac{\partial v^2}{\partial s_0}
			-\frac{\partial v^3}{\partial s_1}\right) &= 0
\end{align*}
and conclude
\begin{align*}
	\frac{\partial v^3}{\partial s_0} &= -a_{01}
	\\
	\frac{\partial v^2}{\partial s_1} &= -a_{10}
	\\
	\frac{\partial v^3}{\partial s_1}-\frac{\partial v^2}{\partial s_0}
	&= a_{00}-a_{11}\,.
\end{align*}
With this additional information, we move on to the block $(*D)$. After
differentiating the (2,2) component w.r.t.~$s_0$ and $s_1$, respectively,
we have
\begin{align*}
	(F_1^2\rho_0'^2+\varepsilon F_0^2\rho_1'^2)\,
	\frac{\del^2u^2}{\del s_0^2} &= 0 \\
	(F_1^2\rho_0'^2+\varepsilon F_0^2\rho_1'^2)\,
	\frac{\del^2u^2}{\del s_0\del s_1} &= 0
\end{align*}
Assuming
\begin{equation}\label{eqn:exceptional.criterion.L}
	F_1^2\rho_0'^2+\varepsilon F_0^2\rho_1'^2\ne0\,,
\end{equation}
we infer
\[
	\frac{\del^2u^2}{\del s_0^2}=0
	\qquad\text{and}\qquad
	\frac{\del^2u^2}{\del s_0\del s_1}=0\,.
\]
Therefore
\[
	v^2(s_0,s_1) = \mu s_0-a_{10}s_1+\nu\qquad
	(\mu,\nu\in\RR)
\]
and then
\[
	v^3(s_0,s_1) = -a_{01}s_0+(\mu+a_{00}-a_{11})s_1+\nu_0\qquad
	(\mu,\nu_0\in\RR)\,.
\]
For this result we have assumed~\eqref{eqn:exceptional.criterion.L}. The next
section is dedicated to the complementary case.

\subsection{The exceptional branch}

According to the lemma, an exceptional situation occurs if the
condition~\eqref{eqn:exceptional.criterion.L} is not satisfied. In this case
we have
\begin{equation}\label{eqn:L.exceptional}
	F_1^2\rho_0'^2+\varepsilon F_0^2\rho_1'^2 = 0\,.
\end{equation}
As $F_i$, $\rho_i$ and $\varepsilon$ are real and non-vanishing, and
$\varepsilon^2=1$, we conclude that~\eqref{eqn:L.exceptional} implies
$$ \varepsilon=-1. $$
The remainder of the section is dedicated to the proof of the main theorem
for the Liouville case.

\subsubsection{Metrics L1}\label{sec:exceptional.L1}
We begin with the metrics of L1 type. We have
\[
	\rho_i=x_i\,,\quad
	F_i = c_i\in\RR.
\]
From the previous discussion we also infer ($t,\tau\in\RR$)
\[
	a_{00}=\frac35t\,,\quad
	a_{01}=0\,,\quad
	a_{11}=-\frac25t\,,\qquad
	a_{10}=\tau\,,\qquad
	\mu = -2t\,.
\]

\paragraph{Exceptional case:}
Condition~\eqref{eqn:L.exceptional} thus becomes
\[
	c_0^2+\varepsilon c_1^2=0\qquad
	(\text{thus $\varepsilon=-1$})\,.
\]
From the equations in block $(*d)$ we obtain, after substitution of the
previously found equations,
\begin{align*}
	\frac{\del v^2}{\del s_1} &= -a_{10} \\
	\frac{\del v^3}{\del s_0} &= -a_{01} \\
	\frac{\del v^2}{\del s_0}-\frac{\del v^3}{\del s_1} &= a_{00}-a_{11}
\end{align*}
Next, substituting everything found so far, the block $(*A)$ yields
\[
	a_{01}=0\,,\quad
	a_{11}=-\frac23a_{00}
\]
Finally, with these solutions, we infer from the block $(*D)$ that
\[
	\frac{\del v^2}{\del s_0} = -\frac{10}{3}a_{00}\,,
\]
and we obtain
\begin{align*}
	v^0 &= -\frac53a_{00}x_0+a_{10} \\
	v^1 &= -\frac53a_{00}x_1+a_{10} \\
	v^2 &= -\frac{10}{3}a_{00}s_0-a_{10}s_1+\nu \\
	v^3 &= -\frac53a_{00}s_1+\nu_0
\end{align*}
and thus a 4-dimensional algebra of symmetries.

\subsubsection{Metrics L2}
Since we are in the exceptional branch, $\varepsilon=-1$, and we find the
condition
$$ c_1^2d_0^2 = c_0^2d_1^2. $$
Solving the remaining equations, we find
$$ a_{01}=0\,,\ a_{11}=-\frac23a_{00}\,,\ \beta=0 $$
and then, solving for the derivatives of $v^j$,
$$ \projalg = \langle
				\del_{s_0}\,,\
				\del_{s_1}\,,\
				\del_{x_0}+\del_{x_1}-2s_0\del_{s_0}-s_1\del_{s_1}\,,\
				\frac{1}{c_0}e^{x_0}\del_{x_0}
					+\frac{1}{c_1}e^{x_1}\del_{x_1}
					-s_1\del_{s_0}
			\rangle $$

\subsubsection{Metrics L3}
Condition~\eqref{eqn:L.exceptional} becomes
$$ \varepsilon c_0^2e^{-3x_0}+c_1^2e^{-3x_1} = 0\,, $$
which cannot be identically satisfied unless $(c_0,c_1)=(0,0)$, which is
impossible.

\subsubsection{Metrics L4}
Using that $\cos(x_0)e^{3\beta x_0}$ and $\cos(x_1)e^{3\beta x_0}$ are
linearly independent functions, Condition~\eqref{eqn:L.exceptional} is
equivalent to the system
\[
	\varepsilon c_0^2=0
	\quad\text{and}\quad
	c_1^2=0\,,
\]
which cannot be satisfied as $g$ would be degenerate.
Therefore the exceptional branch is void in the case L4.

\subsection{Finalising the proof of Theorem~\ref{thm:Liouville.main}}
We return to the main branch, i.e.~we now assume
\[
	 F_1^2\rho_0'^2+\varepsilon F_0^2\rho_1'^2\ne0\,.
\]
Using Lemma~\ref{la:Liouville}, we find that the blocks $(*a)$, $(*b)$,
$(*c)$ and $(*d)$ as well as $(*B)$ and $(*C)$ become redundant.
The block $(*A)$ is diagonal and the block $(*D)$ is symmetric, and these
remaining conditions can be written as equations of the form
\[
\sum_\ell c_\ell F_\ell(x_0,x_1) = 0
\]
where $F_\ell(x_0,x_1)$ are bivariate functions in $x_0,x_1$ and linearly
independent functions. Thus $c_\ell=0$ for all $\ell$.
We solve these remaining conditions separately for each of the metrics of
Liouville type. Each of the following subsections covers one of these metrics.

\subsubsection{Case L1}

The remaining conditions~\eqref{eqn:liouville.cases.algebraic.condition} are
rational polynomials in $x_0,x_1$ and thus lead to polynomial conditions,
where each coefficient has to vanish independently.
We arrive at
$$
	\mu = -\frac{10}{3}a_{00}\,,\quad
	a_{01}=0\,,\quad
	a_{11} = -\frac23a_{00}\,,
$$
and thus, up to a constant factor,
$$
	v = \begin{pmatrix}
			-5a_{00}x_0+3a_{10} \\
			-5a_{00}x_1+3a_{10} \\
			-10a_{00}s_0-3a_{10}s_1+3\nu \\
			-3a_{10}s_1+3\nu_0
		\end{pmatrix}
$$
meaning
$$
	\mathfrak{p}=\langle
					\partial_{s_1}\,,
					\partial_{s_0}\,,
		\partial_{x_0}+\partial_{x_1}-s_1\partial_{s_0}\,,
		x_0\partial_{x_0}+x_1\partial_{x_1}+2s_0\partial_{s_0}+s_1\partial_{s_1}
				\rangle\,.
$$
We note that this algebra coincides with the one found in the exceptional
case, see Section~\ref{sec:exceptional.L1}.

\subsubsection{Case L2}
Since $\beta\ne1$, the remaining
conditions~\eqref{eqn:liouville.cases.algebraic.condition} are equivalent to
\begin{align*}
	a_{01}c_0^2e^{2\beta x_0} &= 0 \\
	a_{01}c_1^2e^{2\beta x_1} &= 0 \\
	a_{10}\beta e^{2\beta x_0} &= 0 \\
	a_{10}\beta e^{2\beta x_1} &= 0 \\
	((5\beta-2)a_{00}-3a_{11})c_0 &= 0 \\
	((5\beta-2)a_{00}-3a_{11})c_1 &= 0
\end{align*}
We solve this system distinguishing two branches:

\noindent\textbf{Case $\beta\ne0$.}
We arrive at
$$
	\mu = -\frac{5}{3}(\beta+2)a_{00}\,,\
	a_{01}=0\,,\
	a_{10}=0\,,\
	a_{11} = \frac13(5\beta-2)a_{00}\,.
$$
This yields
$$
	\mathfrak{p}=\langle
					\partial_{s_1}\,,
					\partial_{s_0}\,,
					\partial_{x_0}+\partial_{x_1}
						-(\beta+2)s_0\partial_{s_0}
						-(2\beta+1)s_1\partial_{s_1}
				\rangle\,.
$$

\noindent\textbf{Case $\beta=0$.}
In this case we arrive at
$$
	\mu = -\frac{10}{3}a_{00}\,,\
	a_{01}=0\,,\
	a_{11} = -\frac23a_{00}\,.
$$
This yields the 4-dimensional algebra
$$
	\mathfrak{p}=\langle
					\partial_{s_0}\,,\
					\partial_{s_1}\,,\
					\frac1{c_0}e^{x_0}\del_{x_0}
						+\frac1{c_1}e^{x_1}\del_{x_1}
						+s_1\del_{s_0}\,,\
					\partial_{x_0}+\partial_{x_1}
						-2s_0\partial_{s_0}
						-s_1\partial_{s_1}
				\rangle\,.
$$

\subsubsection{Case L3}
We obtain the conditions
$$
	\mu = -5a_{00}\,,\quad
	a_{01}=0\,,\quad
	a_{10}=\frac53a_{00}\,,\quad
	a_{11}=a_{00}\,,
$$
and arrive at the 3-dimensional algebra
$$
\mathfrak{p}=\langle
				\partial_{s_1}\,,
				\partial_{s_0}\,,
				\partial_{x_0}+\partial_{x_1}
					-(3s_0+s_1)\partial_{s_0}
					-3s_1\partial_{s_1}
			\rangle\,.
$$

\subsubsection{Case L4}
Solving the remaining conditions
of~\eqref{eqn:liouville.cases.algebraic.condition},
and using that $\cos(x_0)\sin(x_0)$ and $\cos^2(x_0)$
are linearly independent functions, we find the system
\begin{align*}
	(a_{01}+a_{10})\beta-a_{00}+a_{11} &= 0 \\
	(a_{00}-a_{11})\beta+a_{01}+a_{10} &= 0 \\
	3a_{01}\beta-5a_{00} &= 0\,.
\end{align*}
We thus distinguish two cases:

\noindent\textbf{Case $\beta\ne0$.}
If $\beta\ne0$, we arrive at
$$
	\mu = -5a_{00}
	a_{01}=\frac53\frac{a_{00}}{\beta},
	a_{10}=-\frac53\frac{a_{00}}{\beta},
	a_{11}=a_{00}
$$
and thus
$$
	\mathfrak{p}=\langle
					\partial_{s_1}\,,
					\partial_{s_0}\,,
					\partial_{x_0}+\partial_{x_1}
						-(3\beta s_0-s_1)\partial_{s_0}
						-(s_0+3\beta s_1)\partial_{s_1}
				\rangle\,.
$$

\noindent\textbf{Case $\beta=0$.}
The remaining equation of (2) can be solved for
$$ a_{00}=0\,,\quad a_{11}=0\,,\quad a_{01}=-a_{10} $$
and the remaining conditions of (3) then yield
$$ \projalg = \langle
				\del_{s_0}\,,\
				\del_{s_1}\,,\
				\del_{x_0}+\del_{x_1}+s_1\del_{s_0}-s_0\del_{s_1}
			\rangle $$
We observe that this algebra is the same as the one obtained from the one in
the case $\beta\ne0$.

\section{Proof of Theorem~\ref{thm:complex.main} (complex
type)}\label{sec:complex}
Theorem~\ref{thm:complex.main} is a direct corollary of Theorem~\ref{thm:Liouville.main}.
Indeed, we observe a formal analogy of the complex type metrics and the Liouville type metrics, where we formally replace
$\veps=-1$, $x_0\to z$ and $x_1\to\barz$ etc, as well as $c_0=c_1=1$,
$d_0\to\varsigma$ and $d_1\to\bar\varsigma$.
The statement then follows directly by comparison, where we need to replace
the condition $c_0^2d_1^2+\varepsilon c_1^2d_0^2=0$ by
$\varsigma^2-\bar\varsigma^2=0$, which is easily identified as
$\varsigma_0\varsigma_1=0$. Since $\varsigma\ne0$, this corresponds either to
$\varsigma=\varsigma_0\ne0$ with
$\varsigma_0\in\RR$, or to $\varsigma=i\varsigma_1\ne0$ with
$\varsigma_1\in\RR$.

\section{Proof of Theorem~\ref{thm:degenerate.main} (Degenerate
type)}\label{sec:degenerate}

We proceed by again proving an auxialliary result first.
\begin{proposition}\label{prop:degenerate}~

	\noindent
	(i) Any c-projective symmetry of a degenerate metric from
	Fact~\ref{fct:metrics} is of the form
	\[
		v = v^0(x_0)\partial_{x_0}+(\eta x_1+f(s_0,s_1))\partial_{x_1}+u
	\]
	where $u$ is a homothetic vector field of $h$, $\eta\in\RR$.

	\noindent
	(ii) If $u=(u^0,u^1)$ is a homothetic vector field of $h$, $\lie_uh=Ch$,
	satisfying the respective integrability condition in
	Table~\ref{tab:integrability.Dx}, then there are $v^0(x_0)$, $f(s_0,s_1)$
	and $\eta\in\RR$ such that
	\[
		v = v^0(x_0)\partial_{x_0}+(\eta x_1+f(s_0,s_1))\partial_{x_1}+u
	\]
	is a c-projective vector field of a metric $g$ of degenerate type as in
	Fact~\ref{fct:metrics} (see Table~\ref{tab:integrability.Dx} for the
	particular form of $v^0$, $f$ and $\eta$).\\
	The integrability conditions are always satisfied for a Killing vector
	field $u$ of~$h$, $C=0$. Then $f=\text{const}$ and $v$ is Killing
	for~$g$.
\end{proposition}
\begin{table}
	\textbf{Integrability conditions for part~(ii) of
	Proposition~\ref{prop:degenerate}}\medskip

	\begin{tabular}{l|l|l}
		Case & equations for $v^0$ and $v^1$ & integrability condition
		\\
		\toprule
		D1 &
		\begin{minipage}{.4\textwidth}
			$ v^0 = Cx_0+k\qquad(k\in\RR) $\\[.2cm]
			$ \eta = C $ \\[.2cm]
			$ \diff{f}{s_0} = s_0\,\diff{u^1}{s_0} $ \\
			$ \diff{f}{s_1} =
				u^0+u^1\,s_0\tfrac{G'}{G}-Cs_0 $
		\end{minipage}
		& No condition
		\\
		\midrule
		D2a &
		\begin{minipage}{.3\textwidth}
			$ v^0 = k\qquad(k\in\RR) $\\[.2cm]
			$ \eta = 0 $\\[.2cm]
			$ \diff{f}{s_0} = s_0\diff{u^1}{s_0} $ \\
			$ \diff{f}{s_1} = u^0+\tfrac12s_0\,u^1\,\frac{G'}{G} $
		\end{minipage}
		& $C=0$
		\\
		\midrule
		D2b &
		\begin{minipage}{.4\textwidth}
			$ v^0 = -\frac{C}{\beta+2} $\\[.2cm]
			$ \eta = C $ \\[.2cm]
			$ \diff{f}{s_0} = -\frac1{\beta+2}G\,\diff{u^1}{s_0}
				e^{-(\beta+2)s_0} $ \\
			$ \diff{f}{s_1} = \frac{
					CG
					+(\beta+2)G\,u^0
					-G'\,u^1
				}{
					2(\beta+2)\,e^{(\beta+2)s_0}
				}$
		\end{minipage}
		& No condition
		\\
		\midrule
		D3 &
		\begin{minipage}{.4\textwidth}
			$ v^0 = -\tfrac{C}{9}\,c_1e^{-3x_0} $\\[.2cm]
			$ \eta = C $ \\[.2cm]
			$ \diff{f}{s_0} = -\frac13Ge^{-3s_0}\,\diff{u^1}{s_0} $ \\
			$ \diff{f}{s_1} = \frac{
					CG + 3G\,u^0 - G'\,u^1
				}{
					6\,e^{3s_0}
				}$
		\end{minipage}
		& No condition
	\end{tabular}\smallskip

	\caption{Integrability conditions for lifting 2-dimensional homothetic
	vector fields of $h$, $\lie_uh=Ch$, to c-projective vector fields of
	metrics of degenerate type of Fact~\ref{fct:metrics}.}
	\label{tab:integrability.Dx}
\end{table}

\subsection{Proof of the proposition}

\subsubsection{Part~(i)}
The block $(*a)$ again immediately implies $v^0=v^0(x_0)$ and
\[
	v^0 = -\frac{
				a_{01}\rho^2 + (a_{00}-2a_{01}-a_{11})\rho
				-a_{00}+a_{01}-a_{10}+a_{11}
			}{
				\frac{\partial\rho}{\partial x_0}
			}
\]
and, moreover, we obtain
\[
	\frac{\partial v^3}{\partial x_1} = 0\,.
\]
Consider now the blocks $(*b)$ and $(*c)$. We find
\[
	\frac{\partial v^3}{\partial x_0} = 0\quad
	\frac{\partial v^2}{\partial x_0} = 0\quad
	\frac{\partial v^2}{\partial x_1} = 0\,.
\]
We thus confirmed $v^2=v^2(s_0,s_1)$ and $v^3=v^3(s_0,s_1)$.
Moreover, we find, recalling $\tau=f_\tau\,ds_1$ from Fact~\ref{fct:metrics},
\[
	\frac{\partial v^1}{\partial s_0}
	= f_\tau(s_0,s_1)\,\frac{\partial v^3}{\partial s_0}\,.
\]
Now consider the block $(*d)$, which with the previous findings becomes
equivalent to
\[
	a_{11}-a_{00}+a_{01}-a_{10} = 0\,.
\]

From Equation~\eqref{eqn:LvL}, i.e., block $(*b)$ we moreover find
\[
	\diff{v^1}{s_1} = v^2\diff{f_\tau}{s_0}
					+ v^3\diff{f_\tau}{s_1}
					- f_\tau\diff{v^1}{x_1}
					+ f_\tau\diff{v^3}{s_1}\,.
\]
Next, let us take the off-diagonal part of $(*D)$.
We find, using $h=f_0ds_0^2+f_1ds_1^2$,
\[
	f_0(s_0,s_1)\frac{\partial v^2}{\partial s_1}
	+ f_1(s_0,s_1)\frac{\partial v^3}{\partial s_0} = 0\,.
\]
We conclude that there is a function $\phi(s_0,s_1)$ such that
\[
	\frac{\partial v^2}{\partial s_1}=-\phi f_1\qquad
	\frac{\partial v^3}{\partial s_0}=\phi f_0\,.
\]

\noindent\textbf{The case $f_0=f_1=G(s_1)$}
In this case we have
\[
	\frac{\partial v^2}{\partial s_1}=-\phi(s_0,s_1)\qquad
	\frac{\partial v^3}{\partial s_0}=\phi(s_0,s_1)
\]

\noindent\textbf{The case $f_0=\frac1{f_1}=G(s_1)$}
In this case we have
\[
	\frac{\partial v^2}{\partial s_1}=-\phi(s_0,s_1)\qquad
	\frac{\partial v^3}{\partial s_0}=\phi(s_0,s_1)G^2(s_1)
\]

\subsubsection{Part~(ii)}
In order to prove the second part of the claim, we consider each metric of
degenerate type separately.

\noindent\textbf{Case D1}
Taking the block $(*A)$, we infer
\[
	a_{10} = a_{11}
	\quad\text{and}\quad
	\diff{v^1}{x_1} = a_{00} + a_{11}\,.
\]
Resubstituting this into the block $(*D)$, we find
\begin{subequations}
\begin{align}
	\left(a_{00}-a_{10}+2a_{11}-2\diff{v^2}{s_0}\right)G-v^3\diff{G}{s_1} &= 0
	\label{D1.(D).1} \\
	-(a_{00}-a_{11})s_0^2G^2
	+2c_1^2\left(v^3\diff{G}{s_1}
			+\left( \diff{v^2}{s_0}-\diff{v^3}{s_1} \right)G
			\right)x_0
	&=0\,.
	\label{D1.(D).2}
\end{align}
\end{subequations}
Solve~\eqref{D1.(D).1} for $\diff{v^2}{s_0}$,
\begin{equation*}
	\diff{v^2}{s_0} = -\frac12\,\left(
				v^3\frac1G \diff{G}{s_1}-a_{00}+a_{10}-2a_{11}
				\right)\,.
\end{equation*}
The condition~\eqref{D1.(D).2} is a polynomial in $x_0$, in which each
coefficient
needs to vanish independently. Thus, since $G(s_1)\ne0$
\[
	a_{10} = a_{11}
\]
and
\[
	v^3\diff{G}{s_1}
	+ \left(\diff{v^2}{s_0}-\diff{v^3}{s_1}\right)\,G = 0\,.
\]
Replacing $\diff{v^2}{s_0}$, and $a_{10}=a_{11}$,
\[
	\diff{v^3}{s_1}
	= \frac12v^3\frac1G\diff{G}{s_1}
	+\frac12a_{00}+\frac12a_{11}\,.
\]
It remains to consider the block $(*B)$, which is identical to $(*C)$, and
automatically satisfied because of $a_{10} = a_{11}$.
We summarise, with $\hat a=a_{00}+a_{11}$
\begin{align*}
	v^0 &= \hat a x_0-a_{00} \\
	v^1 &= \hat ax_1+w
\end{align*}
where
\begin{align*}
	\diff{w}{s_0} &= s_0\diff{v^3}{s_0} \\
	\diff{w}{s_1} &= s_0\diff{v^3}{s_1}-\hat as_0+v^2
\end{align*}
and $v^2,v^3$ satisfy the conditions of a homothetic vector field on the
2-dimensional block, $C=a_{00}+a_{11}$.

\noindent\textbf{Case D2a}
Taking the block $(*A)$, we infer
\[  a_{10} = -a_{00}+a_{11} \]
and
\[  \diff{v^1}{x_1} = a_{00} + \frac12a_{11}  \]
Resubstituting this into the block (D), we find
\begin{align*}
		\left(2a_{00}+a_{11}-2\diff{v^2}{s_0}\right)G-v^3\diff{G}{s_1} &= 0

		\\
		\left(2a_{00}+a_{11}-2\diff{v^3}{s_1}\right)G
		+v^3\diff{G}{s_1} &= 0\,.

\end{align*}
Solve~\eqref{D1.(D).1} for $\diff{v^2}{s_0}$,
\begin{equation*}
	\diff{v^2}{s_0} = \frac12\left(
						v^3\frac1G \diff{G}{s_1}+2a_{00}+a_{11}
					\right)
\end{equation*}
Similarly,
\[
	\diff{v^3}{s_1}
	= \frac12v^3\frac1G \diff{G}{s_1} + a_{00}+\frac12a_{11}\,.
\]
Again, the blocks $(*B)$ and $(*C)$ are redundant.

We summarise, with $\hat a=a_{00}+\frac12a_{11}$
\begin{align*}
	v^0 &= \frac13(a_{11}-a_{00}) \\
	v^1 &= \hat ax_1+w
\end{align*}
where
\begin{align*}
	\diff{w}{s_0} &= s_0\diff{v^3}{s_0} \\
	\diff{w}{s_1} &= s_0\diff{v^3}{s_1}-\hat as_0+v^2
\end{align*}
and $v^2,v^3$ satisfy the conditions of a homothetic vector field on the
2-dimensional block, $C=2a_{00}+a_{11}$.

\noindent\textbf{Case D2b}
Applying a similar reasoning, we arrive at
\begin{align*}
	v^0 &= -a_{00} \\
	v^1 &= a_{00}(\beta+2)x_1+w
\end{align*}
where
\begin{align*}
	\diff{w}{s_0} &= s_0\diff{v^3}{s_0} \\
	\diff{w}{s_1} &= s_0\diff{v^3}{s_1}-a_{00}(\beta+2)s_0+v^2
\end{align*}
and $v^2,v^3$ satisfy the conditions of a homothetic vector field on the
2-dimensional block (with $C=a_{00}(\beta+2)$).

\noindent\textbf{Case D3}
Analogously, we arrive at
\begin{align*}
	v^0 &= -\frac16(a_{00}-a_{11})c_1e^{-3x_0} \\
	v^1 &= \frac32a_{00}x_1+w(s_0,s_1)
\end{align*}
where
\begin{align*}
	\diff{w}{s_0} &= -\frac13Ge^{-3s_0}\diff{v^3}{s_0} \\
	\diff{w}{s_1} &=\frac1{12}\,\left(
					3(a_{00}-a_{11}+4v^2)G-2v^3\diff{G}{s_1}
				\right)e^{-3s_0}
\end{align*}
and $v^2,v^3$ satisfy the conditions of a homothetic vector field on the
2-dimensional block (with $C=\frac32(a_{00}+a_{11})$).
This concludes the proof of Proposition~\ref{prop:degenerate}.

\subsection{Projective symmetries for the 2-dimensional component}
According to Proposition~\ref{prop:degenerate}, we need to understand
homotheties of 2-dimensional metrics.
So far, we have seen that the projective symmetries of $g$ are extensions of
homothetic symmetries of $h$, even if trivial ones.
In the current section we are therefore going to investigate the local
surfaces with homothetic symmetries.
Looking at Fact~\ref{fct:metrics}, we only have to consider two particular
cases, namely the metrics
\begin{equation}\label{eqn:metrics.f1f}
	G(s_1)ds_0^2+\frac{1}{G(s_1)}ds_1^2
\end{equation}
and the metrics
\begin{equation}\label{eqn:metrics.f}
	e^{Cs_0}G(s_1)(ds_0^2+ds_1^2)\,.
\end{equation}
The metrics \eqref{eqn:metrics.f1f} and \eqref{eqn:metrics.f} admit the
obvious homothetic vector field $\partial_{s_0}$, which is Killing for
\eqref{eqn:metrics.f1f}. For \eqref{eqn:metrics.f} $\del_{s_0}$ is Killing if
$C=0$ and proper homothetic otherwise.
Our goal is to identify those metrics of type \eqref{eqn:metrics.f1f}
or~\eqref{eqn:metrics.f} that admit at least one homothetic vector field in
addition to $\partial_{s_0}$.
We recall that, if a 2-dimensional metric has a 2-dimensional Killing
algebra, it is already of constant curvature (this follows
from~\cite{BMM2008}).

\subsubsection{Metrics \eqref{eqn:metrics.f1f}}
We begin by identifying metrics~\eqref{eqn:metrics.f1f} of
constant curvature.
The following follows by a straightforward computation.
\begin{lemma}\label{la:metric.f1f.constant}
	A metric of type \eqref{eqn:metrics.f1f} has constant curvature if and
	only if
	$$
	\frac{\partial^3G}{\partial s_1^3}=0\,.
	$$

	\noindent(i)
	If $G(s_1)=\kappa s_1^2+\mu_1s_1+\mu_2$ with $\kappa\ne0$ then
	\eqref{eqn:metrics.f1f} has non-zero constant curvature.
	We distinguish three subcases according to the discriminant
	$$ \Delta = \mu_1^2-4\kappa\mu_2 $$
	of $G$ with respect to $s_1$. Thus, a homothetic vector field of $h$ is
	parametrised as
	\begin{align*}
		\Delta&=0 &
		u &= \left(
				\xi_0 +\xi_1\,\left(
					-\kappa s_0^2+4\tfrac{\kappa}{(\diff{G}{s_1})^2}
				\right)
		+\xi_2\,(-2s_0\kappa)
		\right)\del_{s_0}
		+\left(
			\xi_1 s_0\diff{G}{s_1}
			+\xi_2\diff{G}{s_1}
		\right)\del_{s_1}
		\\
		\Delta&>0 &
		u &= \bigg(
		\xi_0 +\xi_1\,\frac{\diff{G}{s_1}}{\sqrt{G}\sqrt{\Delta}}
		\cos(\frac12\sqrt{\Delta}s_0)
		-\xi_2\frac{\diff{G}{s_1}}{\sqrt{G}\sqrt{\Delta}}
		\sin(\frac12\sqrt{\Delta}s_0)
		\bigg)\del_{s_0}
		\\
		&&&\quad
		+\bigg(
		\xi_1\sin(\tfrac12\sqrt{\Delta}s_0)\sqrt{G}
		+\xi_2\cos(\tfrac12\sqrt{\Delta}s_0)\sqrt{G}
		\bigg)\del_{s_1}
		\\
		\Delta&<0 &
		u &= \bigg(
		\xi_0-\xi_1\,\frac{\diff{G}{s_1}}{\sqrt{G}\sqrt{-\Delta}}
		\exp(\frac12\sqrt{-\Delta}s_0)
		+\xi_2\,\frac{\diff{G}{s_1}}{\sqrt{G}\sqrt{-\Delta}}
		\exp(-\frac12\sqrt{-\Delta}s_0)
		\bigg)\del_{s_0}
		\\
		&&&\quad
		+\bigg(
		+\xi_1\exp(\tfrac12\sqrt{-\Delta}s_0)\sqrt{G}
		+\xi_2\exp(-\tfrac12\sqrt{-\Delta}s_0)\sqrt{G}
		\bigg)\del_{s_1}
	\end{align*}

	\noindent(ii)
	If $G(s_1)=\mu_1s_1+\mu_2$  then the homothetic algebra
	is 4-dimensional, and generated by the proper homothetic vector field
	$$ (\mu_1s_1+\mu_2)\partial_{s_1} $$
	and the three Killing vector fields
	\begin{align*}
		\del_{s_0}\,,\qquad
		&\frac{\sin(\tfrac12\mu_1s_0)}{\sqrt{\mu_1s_1+\mu_2}}\,\,\del_{s_0}
		-\cos(\tfrac12\mu_1s_0)\sqrt{\mu_1s_1+\mu_2}\,\,\del_{s_1}\,,
		\\
		&\frac{\cos(\tfrac12\mu_1s_0)}{\sqrt{\mu_1s_1+\mu_2}}\,\,\del_{s_0}
		+\sin(\tfrac12\mu_1s_0)\sqrt{\mu_1s_1+\mu_2}\,\,\del_{s_1}\,.
	\end{align*}
\end{lemma}
Next, we turn to metrics of non-constant Gau{\ss} curvature, aiming to find
those metrics~\eqref{eqn:metrics.f1f} that admit additional homothetic vector
fields, particularly such that are not Killing. Indeed, the existence of two
linearly independent Killing vector fields would already imply that the
metric is of constant curvature \cite{BMM2008}.
The question to be considered thus is for which $G=G(s_1)$ the
metric~\eqref{eqn:metrics.f1f} admits a vector field $u$ such that
\begin{equation}\label{eqn:homothetic.1}
	\mathcal{L}_u\left(Gds_0^2\pm\frac{1}{G}ds_1^2\right)
	=Gds_0^2\pm\frac{1}{G}ds_1^2\,.
\end{equation}
Without losing generality, we may suppose $G>0$, as we work locally and a
constant conformal factor is irrelevant for the question under investigation.
We prove the following lemma.
\begin{lemma}\label{la:non-generic.algebra.f1f}
	Let $Gds_0^2\pm\frac{1}{G}ds_1^2$ be a metric of non-constant curvature
	admitting a homothetic vector field $u$ non-proportional to
	$\partial_{s_0}$.
	Then
	$$
		G=k_3(k_1s_1+k_2)^{\frac{2(k_1+1)}{k_1}}
	$$
	and $u$ is proportional to
	$$
	(k_1+2)s_0\partial_{s_0} - (k_1s_1+k_2)\partial_{s_1}
	$$
	where $k_i\in\mathbb{R}$, $k_1\neq -1$, $k_1\neq -2$ (otherwise the
	metric is of constant curvature).
\end{lemma}
\begin{proof}
	Let $u=u^1\partial_{s_0}+u^2\partial_{s_1}$.
	Equation~\eqref{eqn:homothetic.1} thus yields the system
	\begin{equation}\label{eqn:system.hom.1}
		\left\{
		\begin{array}{l}
			-2G+u^2G_{s_1}+2Gu^1_{s_0}=0
			\\
			G^2u^1_{s_1}\pm u^2_{s_0}=0
			\\
			2Gu^2_{s_1}-u^2G_{s_1}-2G=0
		\end{array}
		\right.
	\end{equation}
	Solving the third equation of~\eqref{eqn:system.hom.1},
	\begin{equation}\label{eqn:X2}
		u^2=\sqrt{G}\left( \int\frac{ds_1}{\sqrt{G}}+F(s_0) \right)
	\end{equation}
	where $F=F(s_0)$ is some univariate function.
	Substituting~\eqref{eqn:X2} into~\eqref{eqn:system.hom.1}, we obtain
	\begin{equation}\label{eqn:system.hom.2}
		\left\{
		\begin{array}{l}
			\pm G^2u^1_{s_1}+\sqrt{G}F_{s_0}=0
			\\
			\sqrt{G}G_{s_1}F + \sqrt{G}G_{s_1} \int\frac{ds_1}{\sqrt{G}} +
			2Gu^1_{s_0}-2G=0
		\end{array}
		\right.
	\end{equation}
	Since $G\neq 0$, we can solve this for $u^1_{s_1}$ and $u^1_{s_0}$,
	respectively. Using the symmetry of second derivatives,
	$\tfrac{\del^2 u^1}{\del s_1\del s_0}
	-\frac{\del^2 u^1}{\del s_0\del s_1}=0$,
	we arrive at
	\begin{equation*}
		F(2GG_{s_1s_1}-G_{s_1}^2)
		= \pm 4F_{s_0s_0} - 2\sqrt{G}G_{s_1}
		-2G\int\frac{ds_1}{\sqrt{G}}G_{s_1s_1}
		+\int\frac{ds_1}{\sqrt{f}}G_{s_1}^2
	\end{equation*}
	In this equation, the right-hand side is a sum of a function
	depending only on $s_0$ and a function depending only on $s_1$. Therefore,
	differentiating the left-hand side with respect to $s_0$ and then with
	respect	to $s_1$ yields zero, and we obtain
	$$
	\frac{\partial^2}{\del s_0\del s_1}
		\left(
			F(s_0)\,\left(
					2G\frac{\del^2G}{\del s_1^2}
					-\left(\diff{G}{s_1}\right)^2
				\right)
		\right)=0\,,
	$$
	implying
	\begin{equation*}
		2G\diff{F}{s_0}\frac{\del^3G}{\del s_1^3}=0\,.
	\end{equation*}
	and thus either $G'''(s_1)=0$ (meaning the metric is of
	constant curvature due to Lemma~\ref{la:metric.f1f.constant}) or
	$F'(s_0)=0$.

	Note that, since $G\neq 0$, combining the first equation
	of~\eqref{eqn:system.hom.2} and the second of \eqref{eqn:system.hom.1}
	yields
	$
	u^1_{s_1}=0\,\,
	\Leftrightarrow u^2_{s_0}=0 \,\,
	\Leftrightarrow \,\,F_{s_0}=0\,.
	$
	We have therefore found: If $u$ is a proper homothetic vector field of
	the metric $Gds_0^2\pm\frac{1}{G}ds_1^2$ of non-constant curvature, then
	\begin{equation}\label{eqn:hom.vec.field.usual}
		u=u^1(s_0)\partial_{s_0}+u^2(s_1)\partial_{s_1}\,.
	\end{equation}
	The claim of the lemma is then proven by substituting
	\eqref{eqn:hom.vec.field.usual} into~\eqref{eqn:system.hom.1} and then
	integrating.
\end{proof}

The homothetic algebra in this latter case is 2-dimensional and generated by
\[
	\partial_{s_0}\quad\text{and}\quad
	(\mu_1+2)s_0\del_{s_0}-(\mu_1s_1+\mu_2)\del_{s_1}\,.
\]

\subsubsection{Metrics \eqref{eqn:metrics.f}}
We analyse metrics of the kind \eqref{eqn:metrics.f} in an analogous manner.
\begin{lemma}
	The metric~\eqref{eqn:metrics.f} has constant curvature if and only
	if $G(s_1)=k_1e^{k_2s_1}$ for constants $k_1,k_2\in\RR$, $k_1\ne0$.
	The constant curvature is then identical to zero and the homothetic
	algebra is 4-dimensional and generated by $\del_{s_0}$ and
	$\mu_1\del_{s_0}+(\beta+2)\del_{s_1}$ as well as
	\begin{align*}

		\exp\left(
				\tfrac{s_0}{2}(\beta+2)-\tfrac12\mu_1s_1
			\right)
		&\sin\left(
				\tfrac12\mu_1s_0+\tfrac12s_1(\beta+2)
			\right)\,\del_{s_0}
		\\
		&-\exp\left(
				\tfrac12s_0(\beta+2)-\tfrac12\mu_1s_1
			\right)
		\cos\left(
				\tfrac12\mu_1s_0+\tfrac12s_1(\beta+2)
			\right)\,\del_{s_1}

		\intertext{and}
		\exp\left(
				\tfrac12s_0(\beta+2)-\tfrac12\mu_1s_1
			\right)
		&\cos\left(
				\mu_1s_0+\tfrac12s_1(\beta+2)
			\right)\,\del_{s_0}
		\\
		&+\exp\left(
				\tfrac12s_0(\beta+2)-\tfrac12\mu_1s_1
			\right)
		\sin\left(
				\tfrac12\mu_1s_0+\tfrac12s_1(\beta+2)
			\right)\,\del_{s_1}\,.
	\end{align*}
\end{lemma}
\begin{proof}
	The scalar curvature of~\eqref{eqn:metrics.f} is
	$$ G^{-3}e^{-\lambda s_0}
		\left(
		\left(\diff{G}{s_1}\right)^2-G\frac{\partial^2G}{\partial s_1^2}
		\right)
	$$
	It is constant if and only if
	\begin{align*}
		\left(
				\left(\diff{G}{s_1}\right)^2
				-G\,\frac{\partial^2G}{\partial s_1^2}
				\right) &= 0
	\end{align*}
	Integrationg for $G$, the claim follows.
\end{proof}

For the case of non-constant curvature we prove the following lemma.
\begin{lemma}\label{la:D2b.D3.2D-hom-algebra}
	Let $h=e^{\lambda s_0}G(s_1)(ds_0^2+ds_1^2)$ ($\lambda\ne0$) be a metric
	of non-constant curvature admitting a homothetic vector field $u$ not
	proportional to $\partial_{s_0}$.
	Then
	$$
		G(s_1)=k_3\sin(k_1s_1+k_2)^{\tfrac{\lambda-2k_1}{k_1}}\,,
	$$
	and $u$ is a linear combination of $\partial_{s_0}$ and
	$$
		w = e^{-k_1s_0}\left(
				\cos(k_1s_1+k_2)\partial_{s_0}
				-\sin(k_1s_1+k_2)\partial_{s_1}
			\right)\,,
	$$
	where $k_i\in\mathbb{R}$, $k_1,k_3\ne0$. In fact, $w$ is Killing for $h$.
\end{lemma}
\begin{proof}
	Since $\lambda\ne0$, $\partial_{s_0}$ is properly homothetic.
	The existence of $u$ as in the hypothesis, implies the existence of a
	Killing vector field. Without loss of generality we therefore assume $u$
	to be Killing.
	The components $u^1,u^2$ of $u$ and the function $G$ then need to satisfy
	the PDE system
	\begin{subequations}\label{eqn:Killing.u.G}
	\begin{align}
		\label{eqn:Killing.u.G.1}
		u^1_{s_0}-u^2_{s_1} &= 0
		\\
		\label{eqn:Killing.u.G.2}
		u^1_{s_1}+u^2_{s_0} &= 0
		\\
		\label{eqn:Killing.u.G.3}
		\lambda\,u^1G+2Gu^2_{s_1}+u^2G_{s_1} &= 0
	\end{align}
	\end{subequations}
	In addition, one can make use of the classification in
	\cite[Theorem~1]{BMM2008}. It is easily seen that only the cases (1a)
	and (2a) of this classification have the properties implied by the
	hypothesis, particularly the commutator of any Killing vector field and
	any homothetic vector fields is proportional to the Killing vector field.
	We conclude that $[u,\del_{s_0}]=cu$ has to hold for some
	constant $c\ne0$, i.e.
	\begin{equation}\label{eqn:commutator.u.delx}
		u^1_{s_0}=-cu^1\,,\qquad u^2_{s_0}=-cu^2
	\end{equation}
	The system composed of~\eqref{eqn:Killing.u.G.1},
	\eqref{eqn:Killing.u.G.2} and~\eqref{eqn:commutator.u.delx} can be
	straightforwardly solved,
	$$
		u^1(s_0,s_1) = \kappa\,e^{-cs_0}\cos(cs_1-\mu)\,,\qquad
		u^2(s_0,s_1) = -\kappa\,e^{-cs_0}\sin(cs_1-\mu)\,.
	$$
	Resubstituting into~\eqref{eqn:Killing.u.G.3}, we have
	$$
		\diff{\ln(G)}{s_1} = (\lambda-2c)\cot(cs_1-\mu)
	$$

	and then
	$$ G = \xi\,\exp\left(\frac{\lambda-2c}{c}\,\ln\sin(cs_1-\mu)\right)
		=\xi\,\sin(cs_1-\mu)^{\frac1c-2}
	$$
	Renaming the integration constants, we arrive at the claim.
\end{proof}

\subsection{Finalising the proof of Theorem~\ref{thm:degenerate.main}}
We finalise the proof by computing the explicit symmetry algebras.

\subsubsection{Case D1}\label{sec:D1.results}
The 2D metric is
\[
	h = G(s_1)ds_0^2+\frac{ds_1^2}{G(s_1)}\,.
\]
We have three scenarios:
\begin{enumerate}
	\item Any homothetic vector field of $h$ is a multiple of the Killing
	vector field $\partial_{s_0}$.
	\item The metric has a $2$-dimensional homothetic algebra. In this
	scenario, w.l.o.g.,
	$$ G(s_1)=\kappa(\mu_1 s_1+\mu_2)^{\frac{2(\mu_1+1)}{\mu_1}} $$
	for $\mu_1,\kappa\ne0$.
	\item $G'''(s_1)=0$, but $G''(s_1)\ne0$, and the metric $h$ is of
	non-zero constant curvature.
\end{enumerate}
Note that if $G''(s_1)=0$, then the metric $h$ is flat. In this case,
however, $g$ is already of constant HSC.

\noindent\textbf{Scenario 1 ($\dim(\homalg(h))=1$).}
We have the homothetic vector fields $u=\xi\partial_{s_0}$ for $h$, and thus obtain
$$ v^0(x_0)=-a_{00} $$
where $a_{00}$ is a free parameter. Moreover, after a straightforward integration, we arrive at
$$ v^1(x_1,s_0,s_1)=\xi s_1+\nu $$
where $\nu\in\RR$. Summarising, the projective algebra of $g$ is 3-dimensional and generated by
$$
	\partial_{x_0}\,,\qquad
	\partial_{x_1}\,,\qquad\text{and}\qquad
	s_1\partial_{x_1}+\partial_{s_0}\,.
$$
We obtain the $2\by2$ matrix $A$ as
$$
A = \begin{pmatrix} a_{00} & a_{00} \\ -a_{00} & -a_{00} \end{pmatrix}\,,
$$
and therefore conclude that $\partial_{x_1}$ and $s_1\partial_{x_1}+\partial_{s_0}$ are Killing vector fields, while
$\del_{x_0}$ is essential as $A_{01}\ne0$.

\noindent\textbf{Scenario 2 ($\dim(\homalg(h))=2$).}
We find
\[
	\phi=0\,,\qquad a_{11}=2-a_{00}
\]
$$
	A = \begin{pmatrix} a_{00} & a_{00} \\ -a_{00} & -a_{00} \end{pmatrix}
$$
and then
\[
	v^1 = 2\xi x_1-\nu_0s_0+\nu\qquad
	v^0 = 2\xi x_0-a_{00}\,.
\]
Therefore, the c-projective vector fields of $g$ are obtained as
\[
	v = (2\xi x_0-a_{00})\partial_{x_0}
		+(2\xi x_1-\nu_0s_1+\nu)\partial_{x_1}
		+(\xi(\mu_1+2)s_0+\nu_0)\partial_{s_0}
		-\xi(\mu_1s_1+\mu_2)\partial_{s_1}\,.
\]
with parameters $\xi,a_{00},\nu_0,\nu\in\RR$.
Written separately, the associated generators of the c-projective symmetry
algebra are
$$
	2x_0\partial_{x_0}+2x_1\partial_{x_1}
			+(\mu_1+2)s_0\partial_{s_0}-(\mu_1s_1+\mu_2)\partial_{s_1}\,,
$$
$$
	\partial_{x_0}\,,\qquad
	\partial_{x_1}\,,\qquad\text{and}\qquad
	s_1\partial_{x_1}+\partial_{s_0}\,.
$$

\noindent\textbf{Scenario 3 ($\dim(\homalg(h))=3$).}
The metric $h$ admits a 3-dimensional homothetic algebra if and only if
$G(s_1)=\kappa s_1^2+\mu_1s_1+\mu_2$.
We obtain $a_{11}=-a_{00}$ and define $\xi=-a_{00}$.

For each subcase, we now compute the induced c-projective vector fields
analogously to the previous scenarios.
We begin with $\Delta=0$. Integrating the PDE system, we find
\[
	v^1 = \xi_1\bigg( \tfrac12 s_0^2(2\kappa s_1+\mu_1)-\frac2{2\kappa
	s_1+\mu_1} \bigg)  +\xi_0 s_1  +\nu
\]
and then
\[
	v = \xi\,\del_{x_0}
		+\bigg(
			\xi_1\big( \tfrac12 s_0^2(2\kappa s_1+\mu_1)-\frac2{2\kappa
			s_1+\mu_1} \big)  +\xi_0 s_1  +\nu
		\bigg)\del_{x_1}
		+u\,.
\]
Next, we compute the subcase $\Delta>0$. We find
\begin{align*}
	v^1 = \xi_0\,s_1
			&+\xi_1\,\sqrt{G}\bigg(
					\tfrac2{\sqrt{\Delta}}\cos(\tfrac12\sqrt{\Delta}s_0)
					+s_0\sin(\tfrac12\sqrt{\Delta}s_0)
				\bigg)
			\\
			&+\xi_2\,\sqrt{G}\bigg(
					s_0\cos(\tfrac12\sqrt{\Delta}s_0)
					-\tfrac2{\sqrt{\Delta}}\sin(\tfrac12\sqrt{\Delta}s_0)
				\bigg)
			+\nu
\end{align*}
and thus
\[
	v = \xi\,\del_{x_0} + v^1\del_{x_1} + u\,.
\]
Finally, for $\Delta<0$,
\[
v^1 = \xi_0\,s_1
		+\xi_1\,\bigg(
				(s_0\Delta-2)\sqrt{\tfrac{G}{-\Delta}}\exp(\tfrac12
				\sqrt{-\Delta} s_0)
		\bigg)
		+\xi_2\,\bigg(
				(s_0\Delta+2)\sqrt{\tfrac{G}{-\Delta}}\exp(-\tfrac12
				\sqrt{-\Delta} s_0)
		\bigg)
		+\nu
\]
and then
\[
	v = \xi\,\del_{x_0}
		+v^1\del_{x_1}
		+u\,.
\]
\bigskip

We recall that $\dim(\homalg(h))\leq3$. In fact, if the homothetic algebra of
$h$ would be larger, its Gau{\ss} curvature would already be zero, and thus
$g$ would have constant HSC due to Theorem~\ref{thm:CHSC}.

\subsubsection{Case D2a}\label{sec:D2a.results}
The metric is as in the case D1 and we continue along the analogous three
cases.
We have four scenarios:
\begin{enumerate}
	\item Any homothetic vector field of $h$ is a multiple of the Killing
	vector field $\partial_{s_0}$.
	\item The metric has a $2$-dimensional homothetic algebra. In this
	scenario, w.l.o.g., $G(s_1)=\kappa(\mu_1 s_1+\mu_2)^{\frac{2(\mu_1+1)}{\mu_1}}$ for
	$\mu_1,\kappa\ne0$.
	\item $G'''(s_1)=0$, but $G''(s_1)\ne0$ and $G''(s_1)\ne\frac9{d_0^2}$, such that the metric $h$ is of
	non-zero constant curvature (but $g$ is of non-constant HSC).
	\item $G''(s_1)=0$ and the metric $h$ is flat.
\end{enumerate}

\noindent\textbf{Scenario 1 ($\dim(\homalg(h))=1$).}
For generic $G$, the only homothetic vector fields of $h$ are ($\xi\in\RR$)
$$ u = \xi\del_{s_0}\,. $$
The integrability condition for $v^1$ is
$$ a_{11} = -2a_{00}\,, $$
and we find
\[
v = -a_{00}\partial_{x_0}
+(\xi s_1+\nu_0)\partial_{x_1}
+\xi\del_{s_0}\,.
\]

\noindent\textbf{Scenario 2 ($\dim(\homalg(h))=2$).}
According to Lemma~\ref{la:non-generic.algebra.f1f},
$$ G=\mu_3(\mu_1s_1+\mu_2)^{\tfrac{2(\mu_1+1)}{\mu_1}} $$
the non-trivial homothetic vector fields of $h$ are multiples of
$$ u=\mu\partial_{s_0}
	+\xi\,\big((\mu_1+2)s_0\del_{s_0}-(\mu_1s_1+\mu_2)\del_{s_1}\big)\,, $$
up to addition of a multiple of $\del_{s_0}$.
From Table~\ref{tab:integrability.Dx}, we infer the integrability condition
$$ \xi=0\,, $$
meaning that only Killing vector fields of $h$ can be extended to symmetries
of the metric~$g$. We find
\[
	v = -a_{00}\partial_{x_0}
		+(\mu s_1+\nu_0)\partial_{x_1}
		+\mu\del_{s_0}
\]
and therefore the c-projective algebra of $g$ is generated by
$$ \partial_{x_0}\,,\quad
	s_1\partial_{x_1}+\del_{s_0}\,,\qquad
	\partial_{x_1}\,. $$

\noindent\textbf{Scenario 3 ($\dim(\homalg(h))=3$).}
We turn to the case when $h$ has a 3-dimensional homothetic algebra. Note
that the homothetic vector fields are analogous to those in Scenario~3 of
case D1. We therefore jump directly to the results.
Using Lemma~\ref{la:metric.f1f.constant}, we obtain $a_{11}=-2a_{00}$ and
abbreviate $\xi=-3a_{00}$.
For $\Delta=0$, we hence find
\[
	v^1 = \xi_0\,s_1
			+\xi_1\,\left(
					\frac12s_0^2(2\kappa s_1+\mu_1)
					-\frac2{2\kappa s_1+\mu_1}
					\right)
			+\nu
\]
and then
\[
	v = \xi\del_{x_0}+v^1\del_{x_1}+u\,.
\]
Next, for $\Delta>0$,
\begin{align*}
	v^1 = \xi_0\,s_1
		&+\xi_1\,\bigg(
			2\tfrac{\sqrt{G}}{\sqrt{\Delta}}\cos(\tfrac12\sqrt{\Delta} s_0)
			+\sqrt{G}s_0\sin(\tfrac12 s_0 \sqrt{\Delta})
		\bigg)
	\\
		&+\xi_2\,\bigg(
			\sqrt{G}s_0\cos(\tfrac12 s_0 \sqrt{\Delta})
			-2\tfrac{\sqrt{G}}{\sqrt{\Delta}}\sin(\tfrac12 s_0 \sqrt{\Delta})
		\bigg)
		+\nu
\end{align*}
and then
\[
	v = \xi\del_{x_0}+v^1\del_{x_1}+u\,.
\]
Finally, for $\Delta<0$,
\begin{align*}
	v^1 = \xi_0\,s_1
			&+\xi_1\,(s_0+\tfrac2{\sqrt{-\Delta}})
					\sqrt{G}\exp(\tfrac12\sqrt{-\Delta}s_0)
		\\
			&+\xi_2\,(s_0-\tfrac2{\sqrt{-\Delta}})
					\sqrt{G}\exp(-\tfrac12\sqrt{-\Delta}s_0)
		+\nu
\end{align*}
and then
\[
	v = \xi\del_{x_0}+v^1\del_{x_1}+u\,.
\]

\noindent\textbf{Scenario 4 ($\dim(\homalg(h))=4$).}
According to Lemma~\ref{la:metric.f1f.constant},
$$ G=\mu_1s_1+\mu_0\,, $$
and the homothetic vector fields of $h$ are parametrised by
\begin{align*}
	 u = \xi_0\del_{s_0}
		&+\xi_1 \bigg(
			\frac{\sin(\tfrac12\mu_1s_0)}{\sqrt{\mu_1s_1+\mu_2}}\del_{s_0}
			-\cos(\tfrac12\mu_1s_0)\sqrt{\mu_1s_1+\mu_2}\del_{s_1}
		\bigg)
		\\
		&+\xi_2 \bigg(
			\frac{\cos(\tfrac12\mu_1s_0)}{\sqrt{\mu_1s_1+\mu_2}}\del_{s_0}
			+\sin(\tfrac12\mu_1s_0)\sqrt{\mu_1s_1+\mu_2}\del_{s_1}
		\bigg)
		+\xi\,\del_{s_1}\,,
\end{align*}
where $\xi$ indicates the properly homothetic contribution and $\xi_i$
parametrise the Killing vector fields.
The integrability condition forces us to put $\xi=0$, and we arrive at
\begin{align*}
 v^1 = \xi_0s_1
	&+\frac{\xi_1}{\mu_1}\,(\mu_1s_0\sin(\tfrac12\mu_1s_0)
					+2\cos(\tfrac12\mu_1s_0))\sqrt{\mu_1s_1+\mu_2}
	\\
	&+\frac{\xi_2}{\mu_1}\,(\mu_1s_0\cos(\tfrac12\mu_1s_0)
					-2\sin(\tfrac12\mu_1s_0))\sqrt{\mu_1s_1+\mu_2}
	+\nu
\end{align*}
where $\nu\in\RR$ is an integration constant. Moreover, we have
$$ v^0 = -a_{00} $$
and thus $g$ admits the 5-dimensional c-projective algebra generated by
$$ \del_{x_0}\,,\quad \del_{x_1}\,,\quad s_1\del_{x_1}+\del_{s_0} $$
as well as
\begin{multline*}
	\big(
		(\mu_1s_0\sin(\tfrac12\mu_1s_0)
		+2\cos(\tfrac12\mu_1s_0))\sqrt{\mu_1s_1+\mu_2}
	\big)\del_{x_1}
	\\
	+\mu_1\,\frac{\cos(\tfrac12\mu_1s_0)}{\sqrt{\mu_1s_1+\mu_2}}\del_{s_0}
	+\mu_1\,\sin(\tfrac12\mu_1s_0)\sqrt{\mu_1s_1+\mu_2}\del_{s_1}
\end{multline*}
and
\begin{multline*}
	 \big(
		(\mu_1s_0\cos(\tfrac12\mu_1s_0)
		-2\sin(\tfrac12\mu_1s_0))\sqrt{\mu_1s_1+\mu_2}
	\big)\del_{x_1}
	\\
	-\mu_1\,\frac{\sin(\tfrac12\mu_1s_0)}{\sqrt{\mu_1s_1+\mu_2}}\del_{s_0}
	+\mu_1\,\cos(\tfrac12\mu_1s_0)\sqrt{\mu_1s_1+\mu_2}\del_{s_1}\,.
\end{multline*}

\subsubsection{Case D2b}\label{sec:D2b.results}
The metric $h$ is
$$ h = e^{-(\beta+2)s_0}G(s_1)(ds_0^2+ds_1^2)\,. $$
We thus have the following three scenarios:
\begin{enumerate}
	\item Any homothetic vector field of $h$ is a multiple of the proper
	homothetic vector field $\partial_{s_0}$.
	\item The metric $h$ admits a 2-dimensional homothetic algebra and is as
	in Lemma~\ref{la:D2b.D3.2D-hom-algebra}.
	\item The metric $h$ has constant curvature. It follows that it is flat.
\end{enumerate}

\noindent\textbf{Scenario 1 ($\dim(\homalg(h))=1$).}
The metric $h$ in this scenario only admits the proper homothetic vector
fields $u = \xi\del_{s_0}$ ($\xi\in\RR$).
Following the same steps as before, we find $v^0=\xi$ and the PDE system for
$v^1(x_1,s_0,s_1)$,
\[
	\diff{v^1}{x_1}+(\beta+2)\xi=0\,,\qquad
	\diff{v^1}{s_0}=0\,,\quad
	\diff{v^1}{s_1}=0\,,
\]
implying
\[
	v^1 = \xi(\beta+2)+\nu\,.
\]
We therefore arrive at
\[
	v = \xi(\del_{x_0}+(\beta+2)x_1\del_{x_1}+\del_{s_0})
		+ \nu\del_{x_1}
\]
and thus the c-projective algebra of $g$ is 2-dimensional.

\noindent\textbf{Scenario 2 ($\dim(\homalg(h))=2$).}
If the homothetic algebra is 2-dimensional, we use
Lemma~\ref{la:D2b.D3.2D-hom-algebra} with $\lambda=-(\beta+2)$ and obtain
$$ v^1 = \kappa\,\frac{k_2k_3}{(k_2+\beta+2)(\beta+2)}\,
				\sin(k_2s_1-k_1)^{-\frac{k_2+\beta+2}{k_2}}\,e^{-s_0(k_2+\beta+2)}
		-\xi\,(\beta+2)x_1+k $$
where $k\in\RR$.
The c-projective algebra of $g$ is therefore 3-dimensional and generated by
$$ \del_{x_1}\,,\quad
	\del_{x_0}
	-(\beta+2)x_1\del_{x_1}
	+\del_{s_0} $$
and
\begin{multline*}
	\frac{k_2k_3}{(k_2+\beta+2)(\beta+2)}\,
		\sin(k_2s_1-k_1)^{-\frac{k_2+\beta+2}{k_2}}\,
		e^{-s_0(k_2+\beta+2)}\del_{x_1}
	\\
	+ e^{-k_2s_0}\cos(k_2s_1-k_1)\del_{s_0}
	- e^{-k_2s_0}\sin(k_2s_1-k_1)\del_{s_1}\,.
\end{multline*}

\noindent\textbf{Scenario 3 ($\dim(\homalg(h))=4$).}
In this case $h$ is flat and we have
$$ G = e^{\mu_1s_1} $$
and the homothetic algebra of $h$ is parametrised by
\begin{align*}
	u &= \xi_0\del_{s_0}
		+\xi_1\,(\mu_1\del_{s_0}+(\beta+2)\del_{s_1})
	\\
	&	+\xi_2\,(\exp(\tfrac{s_0}{2}(\beta+2)-\tfrac12\mu_1s_1)\sin(\tfrac12\mu_1s_0+\tfrac12s_1(\beta+2))\del_{s_0}
	\\
	&\qquad		-\exp(\tfrac12s_0(\beta+2)-\tfrac12\mu_1s_1)\cos(\tfrac12\mu_1s_0+\tfrac12s_1(\beta+2))\del_{s_1})
	\\
	&	+\xi_3\,(\exp(\tfrac12s_0(\beta+2)-\tfrac12\mu_1s_1)\cos(\mu_1s_0+\tfrac12s_1(\beta+2))\del_{s_0}
	\\
	&\qquad		+\exp(\tfrac12s_0(\beta+2)-\tfrac12\mu_1s_1)\sin(\tfrac12\mu_1s_0+\tfrac12s_1(\beta+2))\del_{s_1})
\end{align*}
Using this, we obtain by integration of the PDE system that
\begin{align*}
	v^1 &= -\xi_0\,(\beta+2)x_1 + \nu
	\\
	&\quad +\xi_2\exp(\frac12\mu_1s_1-\frac12s_0(\beta+2))
			\frac{
				 2\mu_1\sin(w)(\beta+2)
				 -\cos(w)\,((\beta+2)^2-\mu_1^2)
			}{
				((\beta+2)^2+\mu_1^2)(\beta+2)
			}
	\\
	&\quad +\xi_3\exp(-\tfrac12(\beta+2)s_0+\tfrac12\mu_1s_1)
			\frac{
				2\mu_1\cos(w)(\beta+2)
				-\sin(w)(\mu_1^2-(\beta+2)^2)
			}{
				((\beta+2)^2+\mu_1^2)(\beta+2)
			}
\end{align*}
where $w=\frac12s_1(\beta+2)+\frac12\mu_1s_0$.
We also find $a_{00}=-\xi_0$ and thus a 4-dimensional c-projective algebra
for $g$,
\[
	v = -\xi_0\del_{x_0}+v^1\del_{x_1}+u\,.
\]

\subsubsection{Case D3}\label{sec:D3.results}
The metric $h$ is
$$ h = e^{-3s_0}G(s_1)(ds_0^2+ds_1^2)\,. $$
We thus have the following three scenarios:
\begin{enumerate}
	\item Any homothetic vector field of $h$ is a multiple of the proper
	homothetic vector field $\partial_{s_0}$.
	\item The metric $h$ admits a 2-dimensional homothetic algebra and is as
	in Lemma~\ref{la:D2b.D3.2D-hom-algebra}.
	\item The metric $h$ has constant curvature. It follows that it is flat.
\end{enumerate}

\noindent\textbf{Scenario 1 ($\dim(\homalg(h))=1$).}
The metric $h$ in this scenario only admits the proper homothetic vector
fields $u = \xi\del_{s_0}$ ($\xi\in\RR$).
Following the same steps as before, we find $v^0=\xi$ and the PDE system for
$v^1(x_1,s_0,s_1)$,
\[
	\diff{v^1}{x_1}+3\xi=0\,,\qquad
	\diff{v^1}{s_0}=0\,,\quad
	\diff{v^1}{s_1}=0\,,
\]
and therefore we arrive at
\[
	v = \xi(\del_{x_0}+3x_1\del_{x_1}+\del_{s_0})
		+ \nu\del_{x_1}
\]
and thus the projective algebra of $g$ is 2-dimensional.

\noindent\textbf{Scenario 2 ($\dim(\homalg(h))=2$).}
If the homothetic algebra is 2-dimensional, we use Lemma~\ref{la:D2b.D3.2D-hom-algebra} and obtain
$$ v^1
  = \kappa\frac{k_1k_3}{3(k_1+3)}\,e^{-(k_1+3)s_0}\,
  			\sin(k_1s_1+k_2)^{-\frac{k_1+3}{k_1}}
  -3x_1
  +k $$
where $k\in\RR$.
The c-projective algebra of $g$ is therefore 3-dimensional and generated by
$$ \del_{x_1}\,,\qquad
	\del_{x_0}-3x_1\del_{x_1}+\del_{s_0} $$
and
$$ \frac{k_1k_3}{3(k_1+3)}\,e^{-(k_1+3)s_0}\,
		\sin(k_1s_1+k_2)^{-\frac{k_1+3}{k_1}}\del_{x_1}
	+ e^{-k_1s_0}\cos(k_1s_1+k_2)\del_{s_0}
	- e^{-k_1s_0}\sin(k_1s_1+k_2)\del_{s_1}\,. $$

\noindent\textbf{Scenario 3 ($\dim(\homalg(h))=4$).}
In this case $h$ is flat and the homothetic algebra is analogous to that of
Scenario~3 of the case D2b, where we formally replace $\beta=1$.
Proceeding as before, we find first
\begin{align*}
	v^1 &= \nu-3\xi_0\,x_1
	\\
	&\quad	+\xi_2\frac{
				(\mu_1^2-9)
				\cos(\tfrac12 \mu_1 s_0+\tfrac32 s_1)
				+6\mu_1\sin(\tfrac12\mu_1 s_0+\tfrac32 s_1)
			}{
				3(\mu_1^2+9)
			}\,
			\exp(\tfrac12\mu_1s_1-\tfrac32 s_0)
	\\
	&\quad	+\xi_3\frac{
			6\mu_1
			\cos(\tfrac12 \mu_1 s_0+\tfrac32 s_1)
			-(\mu_1^2-9)\sin(\tfrac12\mu_1 s_0+\tfrac32 s_1)
		}{
			3(\mu_1^2+9)
		}\,
		\exp(\tfrac12\mu_1s_1-\tfrac32 s_0)
\end{align*}
and then
\[
	v = \xi_0\del_{x_0}+v^1\del_{x_1}+u\,.
\]

This concludes the proof of Theorem~\ref{thm:degenerate.main}.
\bigskip

We have thus obtained the full algebras of c-projective vector fields for
K\"ahler surfaces that admit non-affine c-projective vector fields.
We have, in particular, found that the c-projective vector fields in the case
of degenerate type metrics (D1)--(D3) arise from c-projective vector fields
of a 2-dimensional metric $h$ involved in these metrics, which likely bears
some significance for practical applications. Moreover, this phenomenon
should be expected to arise also in higher dimension, which might be useful
for extending the results obtained here to higher dimensions.

\section{Final remarks}
In the present paper we have obtained the full c-projective algebras for all
K\"ahler surfaces with essential c-projective vector fields, i.e.~for the
K\"ahler metrics described in~\cite{BMMR2015}.
We have seen in Theorem~\ref{thm:CHSC} that, while covering all metrics with
essential c-projective vector fields, the list of metrics in~\cite{BMMR2015}
does still contain metrics different from those of interest. Moreover, one
might argue that the list in~\cite{BMMR2015} describes the metrics only up to
a projective transformation (and still in a non-sharp way). On another level,
one might ask for a description up to isometric transformations. The authors
intend to address this problem in an upcoming paper, which will be
facilitated by the results obtained here.

\section*{Acknowledgements}
The authors thank Vladimir Matveev and Stefan Rosemann for insightful
discussions, remarks and for providing helpful references.
The authors acknowledge support through the project PRIN 2017 ``Real and
Complex Manifolds: Topology, Geometry and holomorphic dynamics'', the project
``Connessioni proiettive, equazioni di Monge-Amp\`ere e sistemi integrabili''
Istituto Nazionale di Alta Matematica (INdAM) and the MIUR grant
``Dipartimenti di Eccellenza 2018-2022 (E11G18000350001)''.
GM acknowledges the ``Finanziamento alla Ricerca
(53{\textbackslash}RBA17MANGIO)''.
GM and AV are members of GNSAGA of INdAM.
JS was a research fellow at INdAM.

redpajama_set_name: RedPajamaArXiv


text:
University of Strathclyde
Location Glasgow, Scotland
Percent International Students 12.1%
Famous For Business, Engineering, Physics
International Fees £13,500-19,100
The University of Strathclyde was formally awarded university status in 1964 and currently has almost 23,000 students enrolled from 100 countries on a single campus in the heart of Glasgow, Scotland. Strathclyde was awarded University of the Year in 2012 and Entrepreneurial University of the Year in 2013 by Times Higher Education.
The University is known both for its business school and close links with industry. Strathclyde is one of only 55 universities in the world that is 'triple accredited' by all three major bodies, and they recently opened another campus near Delhi. Many courses, such as product design and innovation or international business with modern languages, have been specifically designed with industry experts. The engineering faculty is the biggest in Scotland and has the largest electrical power engineering and energy research facility in Europe.
Services for International Students
The International Student Club meets weekly during the academic year and provide an informal forum for students to meet and get to know each other in a relaxed atmosphere. Social events are planned, including day and weekend tours, international food evening events, trips to the theatre and music concerts. Weekly international meetings also provide a great opportunity for students to meet and make valuable contacts and friends.
Pre-sessional and in-sessional English courses are available and students are also entitled to up to four hours per week of free English language tuition and support. Staff are also available to help with other issues such as visas, employment and accommodation.
44th in the Times University Guide 2019
51st in the Guardian University Guide 2020
296th in the QS World University Ranking 2020
The University of Strathclyde Technology & Innovation Centre is transforming Strathclyde's research output and, following the last Research Excellence Framework in 2014, the University moved into the top 20 research intensive universities in the UK. The Physics department is number one in the UK, while Electronic & electrical Engineering is number one in Scotland.
Almost 1,500 students live in the campus village, while a further 400 live in University accommodation within close walking distance of the University. Strathclyde offers halls arranged into flats with a shared kitchen, and all first year undergraduate students are guaranteed a place in University accommodation. A web service is available to connect landlords and students who wish to move off campus after the first year.
The University of Strathclyde is located in Glasgow, Scotland, a hugely vibrant and cosmopolitan city featuring a vibrant arts scene, cutting edge music, and an exciting nightlife. It offers over 20 museums and galleries, and is home to the Scottish Opera, the BBC Scottish Symphony Orchestra, and the Scottish Royal Ballet. An impressive amount of green space is located in the city centre, with 19th century landscaped parks and gardens surrounding students who require a balance to the urban pace of life.
Students can also enjoy the highlands and islands which located just outside of Glasgow, less than an hour away. A shuttle bus is available between the John Anderson campus and the Jordanhill campus.
John Logie Baird (Inventor of television)
Alex Kapranos (Franz Ferdinand)
Ed Bryne (Comedian)
Sinjay Jha (Co-CEO of Motorola)
Tom Hunter (Entrepreneur)

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Can we always find a normal subgroup isomorphic to a Quotient group? I'm not very good with English terms of group theory but here is the question :
$$\forall H\trianglelefteq G \rightarrow \exists H' \trianglelefteq G : {G\over H} \approx H'$$
is above statement always true? or should there be some other constraints?

A: This is not true in general. The smallest counterexample can be found in the quaternion group $Q_8$. There $Q_8/Z(Q_8)$ is isomorphic to the Klein $4$-group $\mathbb{Z}_2 \times \mathbb{Z}_2$, but every subgroup of order $4$ in $Q_8$ is cyclic.
However, if we assume that $G$ is finite and abelian, then the statement is true.

A: Hint: The symmetric group $S_5$ has exactly three normal subgroups: $\{1\}$, $S_5$ and the alternating group $A_5$, which has index 2 in $S_5$.

redpajama_set_name: RedPajamaStackExchange


Calling her Madhuri, he attracts the lower class woman to his rich home but refuses to marry her. A new window will pop up. Smule Are you still there?
Retrieved from " https: Hindi-language films films Indian films s Hindi-language films Films scored by Shankar Jaikishan Films set in country houses Indian film remakes. Send me occasional email updates.
New Password Please choose a new password. Raaj Kumar Hema Malini Raakhee.
Your account has been verified. This recording isn't ready yet. Add genres as tags to help organize the songbook for others. Instructions to verify your account have been sent. Your password has been successfully updated. The discovery that Sumita had a childhood sweetheart, Shekhar Vinod Mehraleads Kumar to plot the downfall of the pair.
Please enable them to sign in. Available on the Appstore.
Available on the Appstore Get it on Google Play. Report Community Violation Copyright Infringement. Her performance received accolades.
Lal Patthar is a Hindi filmproduced by F.
He later meets a much younger woman, Sumita Raakheeand organises a financial deal with her parents and marries her. Instructions to verify your updated email address have been sent. An Error has Occurred An unknown joon has occurred. Views Read Edit View history.


The John Glenn College of Public Affairs offers a minor in Science, Engineering and Public Policy. This minor introduces students to the governmental roles and responsibilities surrounding science, engineering and innovation. The United States spends more than $400 billion in this area of research and development. National security, healthcare, transportation, energy, environment and much more are supported by government funding. In addition to determining spending, topics such as standards, regulations, patents, inventions and medicines are determined by government policy--science greatly influences these outcomes.
The Science, Engineering and Public Policy program minor consists of 12 semester credit hours composed of six credits of core courses from the John Glenn College of Public Affairs and six credits of thematic courses in substantive areas of relating science, engineering and public policy (e.g., Energy and Environmental Policy).
The goal of the thematic specialization is that all students emerge with knowledge about a substantive area of science, technology or engineering policy. Students must select two courses from the thematic areas listed below. They may also petition to count other courses related to science or engineering policy that are not listed here.
** A student may take all three core courses, and one elective in a thematic area if desired.
Credit hours required: A minimum of 12.
Transfer credit hours allowed: A student is permitted to count up to 6 total hours of transfer credit and/or credit by examination toward the minor.
Overlap with the GE permitted: Up to 6 hours.
The minor must be in a different subject than the major.
Each minor completed must contain a minimum of 12 hours apart from the major and/or additional minors (i.e. minors that require more than 12 credit hours may overlap those hours beyond 12 with the major or with another minor).
Minimum C- for each course.
Minimum 2.00 cumulative point-hour ratio.
Coursework graded pass/non-pass do not count as a credit completion.
3 credit hours of courses graded as satisfactory/unsatisfactory are accepted.


Fluorine Chemistry, Volume V focuses on the compositions, reactions, properties, and functions of fluorine compounds. The book first discusses the general chemistry of fluorine compounds and the physical chemistry of fluorocarbons. Vapor pressures; details of interaction of polyatomic molecules; coefficients and equations of state for gases; liquid compressibility; and compressibilities of liquid mixtures are discussed. The text looks at the radiochemistry and radiation chemistry of fluorine. Isotopes of fluorine; useful methods for producing F18 and F20; radiochemical properties and counting of F20; fluorine activities in nuclear reactors; and analytical determinations are described. The selection also provides numerical analysis and tabulated representations. The book also focuses on the industrial and utilitarian features of fluorine chemistry. Historical and economic factors; properties of fluorocarbons; refrigerants and propellants; gaseous dielectrics; fire extinguishing agents; and fluorocarbon surfactants are discussed. The text is a vital source of data for readers wanting to study fluorine compounds.


text:
Would-be MP taking a stand over violence against women
Jenn Selby will run in the general election for the Women's Equality Party in the 'Two Cities' constituency, which has seen both Labour's Steven Saxby and the Conservative Mark Field withdraw from the race
05 November, 2019 — By Jenn Selby
Jenn Selby is the WEP prospective parliamentary candidate for Cities of London and Westminster
THE general election might just have been called this week but in Cities of London and Westminster, a political battle has been raging for months.
Already it has seen current MP Mark Field toppled from power following a tireless campaign from the Women's Equality Party, after he grabbed peaceful protester Janet Barker by the neck at a black-tie event and faced no repercussions.
Our campaign culminated in a protest (pictured) outside his Conservative Party selection meeting, where activists wore red dresses in homage to Barker and cable-tied themselves together physically to stop him entering the luxurious Grosvenor Hotel.
Hours later Field announced that he would be giving up his 20-year career in the House of Commons at the next general election.
He cited political divisions around Brexit as the reason but we believe there were other factors behind his decision.
Thanks to activists – including myself, one of five survivors of male violence standing for the Women's Equality Party in the snap election and his constituency opponent – his position as an MP had become utterly untenable.
Our approach, including two previous red-dress protests, a petition for Field to be recalled, striking billboard campaigns and extensive media coverage, has created a movement that the political elite can no longer ignore.
The red-dress protest outside Conservative Party selection meeting
What we want is simple: for constituents to have the right to sack an MP who is found guilty of harassment, abuse or violence by an independent investigation.
At the moment, under the Recall Act 2015, they do not have this power.
Already we have started negotiating with the main parties to change their ways and fully commit to ending harassment and abuse in Westminster for good, in exchange for us stepping aside.
It is clear that our work has shaken the system in "Two Cities".
Steven Saxby, the Labour candidate in our constituency, has been deselected, suspended and placed under investigation for sexual harassment within the party, as have a number of other Labour figures across the country.
Make no mistake, however, this has not been easy.
As a survivor of male violence, I have exposed the suffering caused by my deeply traumatic experiences to make a point about just how important it is that Westminster leads by example to tackle systemic abuse and misogyny.
If our representatives – the individuals who vote on the very legislation that is meant to tackle this violence – are not held to account for their own violence and abuse, how is anything ever going to change?
Field and Saxby being removed from the race is only the beginning: the Women's Equality Party will not stop until all MPs and candidates accused of harassment and assault are held to account.
• Jenn Selby is the WEP prospective parliamentary candidate for Cities of London and Westminster.
• Editor's note: Steven Saxby has denied all allegations of wrong-doing. Mark Field apologised to Janet Barker but said his actions at Mansion House were due to concerns for security.
Julian Assange case could be deadly for journalism
12 November — By Tim Dawson
We need a real 'equalities' watchdog
04 December — By Raoul Walawalker
People-friendly streets can make Islington a safer, healthier, and fairer place for everyone
Carlton parents' vote has been ignored
07 January — By Shoda Rackal