# Group classification of the two-dimensional magnetogasdynamics equations in
Lagrangian coordinates

S.V. Meleshko<EMAIL_ADDRESS>E.I. Kaptsov<EMAIL_ADDRESS>S.
Moyo<EMAIL_ADDRESS>G.M. Webb<EMAIL_ADDRESS>School of Mathematics,
Institute of Science, Suranaree University of Technology, 30000, Thailand
Research, Innovation and Postgraduate Studies, Stellenbosch University, South
Africa  Center for Space Plasma and Aeronomic Research, The University of
Alabama in Huntsville, Huntsville,
AL 35805, USA

###### Abstract

The present paper is devoted to the group classification of magnetogasdynamics
equations in which dependent variables in Euler coordinates depend on time and
two spatial coordinates. It is assumed that the continuum is inviscid and
nonthermal polytropic gas with infinite electrical conductivity. The equations
are considered in mass Lagrangian coordinates. Use of Lagrangian coordinates
allows reducing number of dependent variables. The analysis presented in this
article gives complete group classification of the studied equations. This
analysis is necessary for constructing invariant solutions and conservation
laws on the base of Noether’s theorem.

###### keywords:

Magnetogasdynamics equations, Lie group of transformations, Group
classification Subject Classification (MSC 2010): 35C99, 76W05

##  1 Introduction

The equations of magnetogasdynamics (MGD) describe motion of a gas under the
action of the internal forces, which consist of the pressure and magnetic
forces. These equations describe phenomena related to plasma flows, for
example, in plasma confinement, as well as physical problems in astrophysics
and fluid metals flows.

The present article considers MGD flows in which dependent variables in Euler
coordinates depend on time and two spatial coordinates. It is assumed that the
continuum is inviscid and non-thermal polytropic gas with infinite electrical
conductivity. For the analysis of equations describing the behavior of such a
continuum, the Lie group analysis method is applied.

Lie point symmetries are an effective tool for analyzing nonlinear
differential equations [1, 2, 3, 4, 5]. They are related with the fundamental
physical principles of the model under consideration and correspond to the
important properties of the differential equations. Finding an admitted Lie
group is one of the first and necessary steps in application of the group
analysis method to partial differential equations. Using found symmetries one
can construct a representation of invariant or partially invariant solution.
The representation of a solution reduces the number of the independent
variables. The group analysis method guarantees that the reduced system of
equations for an invariant solution has fewer independent variables and is
involutive. Admitted symmetry of variational partial differential equations is
a necessary condition for application of Noether’s theorem, which is used for
deriving conservation laws.

Applications of the group analysis method for different versions of MGD
equations have been considered in many publications. For example, the case of
the finite conductivity was investigated in [6, 7]. The case of the infinite
conductivity was examined in [8, 9]. Invariant solutions were considered in
[10, 11, 12, 13, 14, 15, 16]. Comprehensive analysis of MGD equations in
Eulerian and Lagrangian coordinates with plain and cylindrical symmetries were
given in [17, 18].

The present paper is devoted to the group classification of the MGD equations,
where all dependent variables in Eulerian coordinates depend on time and two
spatial coordinates111Such solutions can also be three-dimensional.. The study
is performed in mass Lagrangian coordinates. The transition to mass Lagrangian
coordinates makes it possible to solve four MGD equations. As a result of this
solving, four arbitrary functions of the mass Lagrangian coordinates are
obtained. In group analysis, these functions are called arbitrary elements.
The presence of arbitrary elements requires a group classification, which
consists of finding all Lie groups admitted by a system of partial
differential equations [2, 3, 4]. In practice, groups are represented by their
generators. The generators admitted for all arbitrary elements form the kernel
of the admitted Lie algebras. The group classification represents all non-
equivalent extensions of the kernel and the corresponding concrete forms of
arbitrary elements, where the equivalence is considered with respect to
equivalence transformations that preserve the structure of the equations, but
can change arbitrary elements.

The paper is organized as follows. The next section provides MGD equations in
mass Lagrangian coordinates. Derivation of the equations in Lagrangian
coordinates, when the dependent variables in Eulerian coordinates depend on
time and two independent space variables. Section 4 provides equivalence
transformations, which are used for simplification arbitrary elements.
Sections 5 and 7 give the group classifications of nonisentropic and
isentropic solutions when $b_{01}^{2}+b_{02}^{2}\neq 0$. Sections 6 and 8 are
devoted to the group classifications of nonisentropic and isentropic solutions
when $b_{01}^{2}+b_{02}^{2}=0$. Conclusions are stated in the last Section.

##  2 Magnetogasdynamics equations in mass Lagrangian coordinates

The magnetogasdynamics equations of an ideal perfect polytropic gas can be
written in the following form [19, 15]

$\begin{array}[]{c}D\rho+\rho\,div\,{\bf u}=0,\\\ D{\bf
u}+\rho^{-1}\nabla(p+\frac{1}{2}{\bf H}^{2})-\rho^{-1}({\bf H}\cdot\nabla){\bf
H}=0,\\\ D{\bf H}+{\bf H}\,div\,{\bf u}-({\bf H}\cdot\nabla){\bf
u}=0,\,\,\,div\,{\bf H}=0,\\\ DS=0,\end{array}$ (2.1)

where $\rho$, ${\bf u}$, $p$, $S$, and ${\bf H}$ correspond to the gas
density, fluid velocity, pressure, entropy and magnetic induction,
respectively, and $\gamma$ is the polytropic exponent,

$D=\partial_{t}+{\bf u}\cdot\nabla,\,\,{\bf H}=(H_{1},H_{2},H_{3}),\,\,{\bf
u}=(u_{1},u_{2},u_{3}),\,\,\,\boldsymbol{x}=(x_{1},x_{2},x_{3}).$

The magnetic field strength $\boldsymbol{{\bf H}}$ and magnetic field
induction ${\bf B}$ are related by the equation ${\bf B}=\sqrt{\mu_{0}}{\bf
H}$, where $\mu_{0}$ is the magnetic permeability. The pressure $p$, the
density $\rho$ and the entropy $S$ are related by the state equation
$p=A(S)\rho^{\gamma}$, where $A(S)=Re^{(S-S_{0})/c_{v}}$, $R$ is the gas
constant, $c_{v}$ is the dimensionless specific heat capacity at constant
volume, and $S_{0}$ is constant.

In coordinate form equations (2.1) become

$\displaystyle\rho_{t}+u_{i}\rho_{x_{i}}+\rho u_{ix_{i}}=0,$ (2.2a)
$\displaystyle\rho(u_{jt}+u_{i}u_{jx_{i}})+H_{i}H_{ix_{j}}-H_{i}H_{jx_{i}}+p_{x_{j}}=0,\,\,\,(j=1,2,3),$
(2.2b) $\displaystyle
H_{jt}+u_{i}H_{jx_{i}}+H_{j}u_{ix_{i}}-H_{i}u_{jx_{i}}=0,\,\,\,(j=1,2,3),$
(2.2c) $\displaystyle H_{ix_{i}}=0,$ (2.2d) $\displaystyle
S_{t}+u_{i}S_{x_{i}}=0,$ (2.2e)

where the energy equation is rewritten. Here summation with respect to a
repeated index is assumed.

The mass Lagrangian coordinates are introduced by the relations

$\rho=J^{-1},\,\,\,\varphi_{it}(t,\xi)=u_{i}(t,\varphi(t,\xi)),$ (2.3)

where

$\xi=(\xi_{1},\xi_{2},\xi_{3}),\,\,\,\varphi=(\varphi_{1},\varphi_{2},\varphi_{3}),\,\,\,J=\det\left(\frac{\partial\varphi}{\partial\xi}\right),\,\,\,T=\frac{\partial\varphi}{\partial\xi}=\left(\begin{array}[]{ccc}\varphi_{1,1}&\varphi_{2,1}&\varphi_{3,1}\\\
\varphi_{1,2}&\varphi_{2,2}&\varphi_{3,2}\\\
\varphi_{1,3}&\varphi_{2,3}&\varphi_{3,3}\end{array}\right),$

and ${\displaystyle\varphi_{i,j}=\frac{\partial\varphi_{i}}{\partial x_{j}}}$.

In mass Lagrangian coordinates the conservation law of mass (2.2a) becomes
identical and equation (2.2e) gives that $S=S_{0}(\xi)$, where $S_{0}(\xi)$ is
an arbitrary function.

For the sake of completeness we provide here the transition of equations (2.2)
to mass Lagrangian coordinates [15]. Let $A=JT^{-1}$, then

$A_{ik}T_{kl}=J\delta_{il}.$

Noting that
${\displaystyle\frac{\partial}{\partial\xi_{j}}=\varphi_{i,j}\frac{\partial}{\partial
x_{i}}}$, the operators

$\frac{\partial}{\partial x}=\left(\begin{array}[]{c}\frac{\partial}{\partial
x_{1}}\\\ \frac{\partial}{\partial x_{2}}\\\ \frac{\partial}{\partial
x_{3}}\end{array}\right),\,\,\,\frac{\partial}{\partial\xi}=\left(\begin{array}[]{c}\frac{\partial}{\partial\xi_{1}}\\\
\frac{\partial}{\partial\xi_{2}}\\\
\frac{\partial}{\partial\xi_{3}}\end{array}\right),$

can be represented as follows

$\frac{\partial}{\partial\xi}=T\frac{\partial}{\partial
x},\,\,\,\frac{\partial}{\partial x}=J^{-1}A\frac{\partial}{\partial\xi}.$

Gauss’ law (2.2d) gives

$JH_{ix_{i}}=A_{ik}H_{i\xi_{k}}=0.$ (2.4)

Direct calculations show that

$\frac{\partial}{\partial\xi_{k}}(A_{ik})=0,\,\,\,\forall i.$ (2.5)

The latter leads to the relations

$A_{ik}H_{j\xi_{k}}=(A_{ik}H_{j})_{\xi_{k}},\,\,\,\,A_{ik}p_{\xi_{k}}=(A_{ik}p)_{\xi_{k}},\,\,\,\forall
i,\,j.$

Using these relations, the part of momentum equations (2.2b) in Lagrangian
coordinates become

$H_{i}H_{ix_{j}}-H_{i}H_{jx_{i}}+p_{x_{j}}=J^{-1}(H_{i}A_{jk}H_{i\xi{}_{k}}-H_{i}A_{ik}H_{j\xi_{k}}+A_{jk}p_{\xi_{k}})=$
$=J^{-1}(\frac{1}{2}A_{jk}\frac{\partial
H^{2}}{\partial\xi{}_{k}}-H_{i}(A_{ik}H_{j})_{\xi_{k}}+A_{jk}p_{\xi_{k}})=$
$=J^{-1}(\frac{\partial}{\partial\xi{}_{k}}(\frac{1}{2}A_{jk}H^{2}+A_{jk}p)-H_{i}(A_{ik}H_{j})_{\xi_{k}}).$

By virtue of Gauss’ law (2.4), one derives that

$H_{i}(A_{ik}H_{j})_{\xi_{k}}=(H_{i}A_{ik}H_{j})_{\xi_{k}}-A_{ik}H_{i\xi_{k}}H_{j}=(H_{i}A_{ik}H_{j})_{\xi_{k}}.$

Hence,

$H_{i}H_{ix_{j}}-H_{i}H_{jx_{i}}+p_{x_{j}}=J^{-1}\frac{\partial}{\partial\xi{}_{k}}\left(A_{jk}(\frac{1}{2}H^{2}+p)-H_{i}A_{ik}H_{j}\right)=$
$=J^{-1}\frac{\partial}{\partial\xi{}_{k}}\left(\delta_{ij}A_{ik}(\frac{1}{2}H^{2}+p)-H_{i}A_{ik}H_{j}\right)=$
$=J^{-1}\frac{\partial}{\partial\xi{}_{k}}\left(A_{ik}\left(\delta_{ij}(\frac{1}{2}H^{2}+p)-H_{i}H_{j}\right)\right)$

Then the momentum equations in Lagrangian coordinates have the form

$\frac{\partial^{2}\varphi_{j}}{\partial
t^{2}}+\frac{\partial}{\partial\xi{}_{k}}\left(A_{ik}\left(\delta_{ij}(\frac{1}{2}H^{2}+p)-H_{i}H_{j}\right)\right)=0.$

Faraday’s equations (2.2c) in Lagrangian coordinates reduces as follows. Let
$\boldsymbol{\boldsymbol{b}}=\rho^{-1}\boldsymbol{{\bf H}}$, then using the
conservation law of mass and Faraday’s equations, one obtains

$\frac{db_{j}}{dt}=-\rho^{-2}\frac{d\rho}{dt}H_{j}+\rho^{-1}\frac{dH_{j}}{dt}=\rho^{-1}H_{i}u_{jx_{i}}=b_{i}u_{jx_{i}}.$

Introducing the vector $\boldsymbol{\boldsymbol{b}}_{0}$ such that
$\boldsymbol{\boldsymbol{b}}=T\boldsymbol{\boldsymbol{b}}_{0}$, one derives

$\frac{\partial b_{j}}{\partial
t}-J^{-1}b_{i}A_{ik}u_{j\xi_{k}}=\frac{\partial b_{0\alpha}}{\partial
t}\varphi_{j,\alpha}+b_{0\alpha}u_{j\xi_{\alpha}}-b_{0\alpha}(J^{-1}T_{\alpha
i}A_{ik})u_{j\xi_{k}}=\frac{\partial b_{0\alpha}}{\partial
t}\varphi_{j,\alpha}=0.$

The latter gives that

$\frac{\partial b_{0\alpha}}{\partial t}=0,\,\,\,\forall\alpha.$

Hence, similar to the entropy, one integrates the Faraday’s equation
$\boldsymbol{\boldsymbol{b}}_{0}=\boldsymbol{\boldsymbol{b}}_{0}(\xi)$, where
$\boldsymbol{\boldsymbol{b}}_{0}(\xi)=(b_{01}(\xi),b_{02}(\xi),b_{03}(\xi))$
is an arbitrary vector function of $\xi$. Gauss’s equation (2.4) reduces as
follows

$H_{ix_{i}}=(\rho b_{i})_{x_{i}}=(\rho
b_{0\alpha}\varphi_{i,\alpha})_{x_{i}}=J^{-1}A_{ik}(\rho
b_{0\alpha}\varphi_{i,\alpha})_{\xi_{k}}=J^{-1}(J^{-1}A_{ik}b_{0\alpha}\varphi_{i,\alpha})_{\xi_{k}}$
$=J^{-1}(J^{-1}T_{\alpha
i}A_{ik}b_{0\alpha})_{\xi_{k}}=J^{-1}(b_{0k})_{\xi_{k}}=0.$

Therefore, in mass Lagrangian coordinates equations (2.2) reduce to the
equations

$\frac{\partial^{2}\varphi_{j}}{\partial
t^{2}}+\frac{\partial}{\partial\xi{}_{k}}\left(A_{ik}\left(\delta_{ij}(\frac{1}{2}H^{2}+p)-H_{i}H_{j}\right)\right)=0,\,\,\,(j=1,2,3),\,\,\,\frac{\partial}{\partial\xi{}_{k}}b_{0k}=0,$
(2.6) $S=S(\xi),\,\,\,\boldsymbol{b}_{0}=\boldsymbol{b}_{0}(\xi),$

where

$H_{i}=J^{-1}b_{0\alpha}\varphi_{i,\alpha},\,\,\,H^{2}=J^{-2}b_{0\alpha}b_{0\beta}\varphi_{i,\alpha}\varphi_{i,\beta}.$

##  3 Equations (2.1) with two independent space variables in Lagrangian
coordinates

We study the case, where all dependent functions in Eulerian coordinates only
depend on two space variables $x_{1}$ and $x_{2}$. From equations (2.3) one
obtains the Cauchy problem222Here the Lagrangian space variables $\xi_{i}$ are
considered before the transition to the mass Lagrangian coordinates.

$(\varphi_{1,3})_{t}=u_{1x_{1}}\varphi_{1,3}+u_{1x_{2}}\varphi_{2,3},\,\,\,\varphi_{1,3}(0,\xi_{1},\xi_{2},\xi_{3})=0,$
$(\varphi_{2,3})_{t}=u_{2x_{1}}\varphi_{1,3}+u_{2x_{2}}\varphi_{2,3},\,\,\,\varphi_{2,3}(0,\xi_{1},\xi_{2},\xi_{3})=0.$

For sufficiently smooth functions $\boldsymbol{u}(t,\boldsymbol{x})$ the
latter Cauchy problem has unique solution $\varphi_{i,3}=0,\,\,\,(i=1,2)$ that
means

$\varphi_{1}=\varphi(t,\xi_{1},\xi_{2}),\,\,\,\varphi_{2}=\zeta(t,\xi_{1},\xi_{2}).$

In this case the transition from Lagrangian coordinates to the mass Lagrangian
coordinates can be done such that
$\varphi_{3}(0,\xi_{1},\xi_{2},\xi_{3})=\xi_{3}$. Hence, because of the
uniqueness of a solution of the Cauchy problem

$(\varphi_{3,3})_{t}=u_{3x_{1}}\varphi_{\xi_{3}}+u_{3x_{2}}\zeta_{\xi_{3}}=0,\,\,\,\varphi_{3,3}(0,\xi_{1},\xi_{2},\xi_{3})=1,$

on gets $\varphi_{3}=\xi_{3}+\chi(t,\xi_{1},\xi_{2})$. Further we use the
notations $\,\xi_{1}=\xi,\,\,\,\xi_{2}=\eta$. Thus, one has

$T=\frac{\partial\varphi}{\partial\xi}=\left(\begin{array}[]{ccc}\varphi_{\xi}&\zeta_{\xi}&\chi_{\xi}\\\
\varphi_{\eta}&\zeta_{\eta}&\chi_{\eta}\\\
0&0&1\end{array}\right),\,\,\,A=\left(-\begin{array}[]{ccc}\zeta_{\eta}&-\zeta_{\xi}&\chi_{\eta}\zeta_{\xi}-\chi_{\xi}\zeta_{\eta}\\\
\varphi_{\eta}&\varphi_{\xi}&\chi_{\xi}\varphi_{\eta}-\chi_{\eta}\varphi_{\xi}\\\
0&0&\varphi_{\xi}\zeta_{\eta}-\varphi_{\eta}\zeta_{\xi}\end{array}\right),\,\,\,J=\varphi_{\xi}\zeta_{\eta}-\varphi_{\eta}\zeta_{\xi},$
$b_{1}=b_{01}\varphi_{\xi}+b_{02}\varphi_{\eta},\,\,\,b_{2}=b_{01}\zeta_{\xi}+b_{02}\zeta_{\eta},$

$b_{3}=b_{01}\chi_{\xi}+b_{02}\chi_{\eta}+b_{03}.$

The latter relations provide that $b_{0i}=b_{0i}(\xi,\eta),\,\,\,(i=1,2,3)$.
As all functions only depend on $\xi$ and $\eta$, and the coefficients
$A_{31}=0$ and $A_{32}=0$, then equations (2.6) become

$\displaystyle{\displaystyle\frac{\partial^{2}\varphi_{j}}{\partial
t^{2}}+\sum_{k=1}^{2}\sum_{i=1}^{2}\frac{\partial}{\partial\xi{}_{k}}\left(A_{ik}\left(\delta_{ij}(\frac{1}{2}H^{2}+p)-H_{i}H_{j}\right)\right)=0,\,\,\,(j=1,2),}$
(3.1a) $\displaystyle{\displaystyle\frac{\partial^{2}\chi}{\partial
t^{2}}-\sum_{k=1}^{2}\sum_{i=1}^{2}\frac{\partial}{\partial\xi_{k}}\left(A_{ik}H_{i}H_{3}\right)=0,}$
(3.1b)

where

$H_{1}=J^{-1}(b_{01}\varphi_{\xi}+b_{02}\varphi_{\eta}),\,\,\,H_{2}=J^{-1}(b_{01}\zeta_{\xi}+b_{02}\zeta_{\eta}),$
$H_{3}=J^{-1}(b_{01}\chi_{\xi}+b_{02}\chi_{\eta}+b_{03}),\,\,\,H^{2}=H_{1}^{2}+H_{2}^{2}+H_{3}^{2},$

and

$S=S(\xi,\eta),\,\,\,\boldsymbol{b}_{0}=(b_{01}(\xi,\eta),b_{02}(\xi,\eta),b_{03}(\xi,\eta))$

are arbitrary functions such that

$\frac{\partial}{\partial\xi}b_{01}+\frac{\partial}{\partial\eta}b_{02}=0.$
(3.2)

##  4 Equivalence transformations

The class of equations (3.1) is parameterized by arbitrary elements
$S(\xi,\eta)$, $b_{0i}(\xi,\eta)$, ($i=1,2,3$). The first step of the group
classification of the class of equations of form (3.1) consists of describing
the equivalence among the equations of this class. The group classification is
considered with respect to these equivalence transformations.

Direct calculations show that the transformations corresponding to the
generators

$X_{1}^{e}=\frac{\partial}{\partial\xi},\,\,\,X_{2}^{e}=\frac{\partial}{\partial\eta},\,\,\,X_{3}^{e}=\frac{\partial}{\partial\varphi},\,\,\,X_{4}^{e}=\frac{\partial}{\partial\zeta},\,\,\,X_{5}^{e}=\frac{\partial}{\partial\chi},\,\,\,X_{6}^{e}=\frac{\partial}{\partial
t},$
$X_{7}^{e}=t\frac{\partial}{\partial\varphi},\,\,\,X_{8}^{e}=t\frac{\partial}{\partial\zeta},\,\,\,X_{9}^{e}=t\frac{\partial}{\partial\chi},\,\,\,X_{10}^{e}=\zeta\frac{\partial}{\partial\varphi}-\varphi\frac{\partial}{\partial\zeta},$
$X_{11}^{e}=t\frac{\partial}{\partial
t}+\xi\frac{\partial}{\partial\xi}+\eta\frac{\partial}{\partial\eta}+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi},$
$X_{12}^{e}=t\frac{\partial}{\partial
t}+2\xi\frac{\partial}{\partial\xi}+2\eta\frac{\partial}{\partial\eta}+4(1-\gamma)S\frac{\partial}{\partial
S}-2b_{03}\frac{\partial}{\partial b_{03}},$
$X_{13}^{e}=-t\frac{\partial}{\partial t}+2S\frac{\partial}{\partial
S}+b_{01}\frac{\partial}{\partial b_{01}}+b_{02}\frac{\partial}{\partial
b_{02}}+b_{03}\frac{\partial}{\partial b_{03}},$
$X_{\lambda_{1}}^{e}=\lambda_{1}(\xi,\eta)\frac{\partial}{\partial\chi},\,\,\,X_{\lambda_{2}}^{e}=t\lambda_{2}(\xi,\eta)\frac{\partial}{\partial\chi}.$

do not change the structure of equations (3.1) and (3.2). Here the generators
$X_{i}^{e},\,\,\,(i=3,4,...11)$ are inherited by equations in Eulerian
coordinates (2.2), where $X_{3}^{e}$, $X_{4}^{e}$, $X_{5}^{e}$ correspond to
the shifts with respect to $x_{i},\,\,\,(i=1,2,3)$; $X_{6}^{e}$, $X_{7}^{e}$,
$X_{8}^{e}$ correspond to the Galilean boosts; $X_{10}^{e}$ correspond to the
rotation. The generator $X_{\lambda_{1}}^{e}$ allows adding a function
$\lambda_{1}(\xi,\eta)$ to $\chi$. In particular, for given
$b_{0i}(\xi,\eta),\,\,\,(i=1,2,3)$ such that $b_{01}^{2}+b_{02}^{2}\neq 0$,
choosing a function $\lambda_{1}(\xi,\eta)$ satisfying the condition

$b_{01}\lambda_{1\xi}+b_{02}\lambda_{1\eta}+b_{03}=0,$

one can assume that after the transformation $b_{03}=0$. Indeed, for
$\chi=\tilde{\chi}+\lambda_{1}$ one derives that

$b_{3}=b_{01}\tilde{\chi}_{\xi}+b_{02}\tilde{\chi}_{\eta}+b_{01}\lambda_{1\xi}+b_{02}\lambda_{1\eta}+b_{03}=b_{01}\tilde{\chi}_{\xi}+b_{02}\tilde{\chi}_{\eta}.$

There are also two involutions

$\begin{array}[]{c}E_{1}:\,\,\,t\rightarrow-t,\\\
E_{2}:\,\,\,(\xi,\eta,\varphi,\zeta,\chi)\rightarrow-(\xi,\eta,\varphi,\zeta,\chi),\end{array}$

where only changeable variables are presented.

The admitted generator $X$ is sought in the form

$X=\xi^{\xi}\frac{\partial}{\partial\xi}+\xi^{\eta}\frac{\partial}{\partial\eta}+\xi^{t}\frac{\partial}{\partial
t}+\zeta^{\varphi}\frac{\partial}{\partial\varphi}+\zeta^{\zeta}\frac{\partial}{\partial\zeta}+\zeta^{\chi}\frac{\partial}{\partial\chi},$

where all coefficients of the generator $X$ depend on
$(t,\xi,\eta,\varphi,\zeta,\chi)$. The determining equations [2] are obtained
by applying the prolongation of the generator $X$ to the left-hand side of
equations (3.1):

$XF_{|(\ref{eq:2Dequations})}=0,$

where $F$ is the left-hand side of equations (3.1), and
$|(\ref{eq:2Dequations})$ means to consider $XF$ on the manifold defined by
equations (3.1).

The analysis of the determining equations depend on the relations between the
entropy $S(\xi,\eta)$ and the vector $\boldsymbol{b}_{0}(\xi,\eta)$. It breaks
down into several cases. Globally, according to the equivalence
transformations corresponding to the generator $X_{f}^{e}$, it decomposes into
$b_{01}^{2}+b_{02}^{2}\neq 0$ and $b_{01}^{2}+b_{02}^{2}=0$, and each of these
cases is divided into non-isentropic and isentropic solutions.

##  5 Nonisentropic case with $b_{01}^{2}+b_{02}^{2}\neq 0$

The general solution of Gauss’ equation (3.2) can be written as

$b_{01}=\psi_{\eta},\,\,\,b_{02}=-\psi_{\xi},$

where $\psi=\psi(\xi,\eta)$. One also can assume that $\psi_{\eta}\neq 0$. By
virtue of the equivalence transformation corresponding to the generator
$X_{f}^{e}$ it can be considered that $b_{03}=0$.

Partially solving the determining equations one derives that
$\xi^{\xi}=\xi^{\xi}(\xi,\eta)$, $\xi^{\eta}=\xi^{\eta}(\xi,\eta)$, and

$\zeta^{\varphi}=(k_{2}+\frac{1}{2}k_{5})\varphi+k_{1}\zeta+k_{7}t+k_{8},\,\,\,\zeta^{\zeta}=(k_{2}+\frac{1}{2}k_{5})\zeta-
k_{1}\varphi+k_{9}t+k_{10},$
$\zeta^{\chi}=(k_{2}+\frac{1}{2}k_{5})\chi+t\lambda_{1}+\lambda_{2},\,\,\,\xi^{t}=k_{5}t+k_{6},$

where $k_{i}$ are constant, and $\lambda_{i}=\lambda_{i}(\psi)$ are arbitrary
functions. The remaining equations are

$\begin{array}[]{c}2(1-\gamma)\xi_{\eta}^{\xi}\psi_{\eta}Sj_{1}+\xi^{\xi}\left(2S(1-\gamma)(j_{1\eta}\psi_{\eta}+g(S_{\eta\eta}\psi_{\eta}-\psi_{\eta\eta}S_{\eta})-(2\gamma-1)\psi_{\eta}S_{\xi}S_{\eta}\right)\\\
+\xi^{\eta}\left(2(1-\gamma)S(S_{\eta\eta}\psi_{\eta}-\psi_{\eta\eta}S_{\eta})+(2\gamma-1)\psi_{\eta}S_{\eta}{}^{2}\right)+(2k_{2}(1-2\gamma)+k_{5})S\psi_{\eta}S_{\eta}=0,\end{array}$
(5.1)
$\begin{array}[]{c}\xi^{\xi}\left(2(1-\gamma)S(j_{1}g\psi_{\eta\eta}+\psi_{\eta}j_{1\xi})+j_{1}\psi_{\eta}((2\gamma-1)S_{\xi}-2(\gamma-1)Sg_{\eta})\right)\\\
+\xi^{\eta}\left(2(1-\gamma)S(\psi_{\eta\eta}j_{1}+j_{1\eta}\psi_{\eta})+(2\gamma-1)S_{\eta}\psi_{\eta}j_{1}\right)-(2k_{2}(\gamma+2)+k_{5}(2\gamma+3)Sj_{1}\psi_{\eta}=0,\end{array}$
(5.2)
$\begin{array}[]{c}\xi_{\xi}^{\eta}=-g^{2}\xi_{\eta}^{\xi}-\xi^{\xi}\left(2gg_{\eta}+2\frac{\psi_{\eta\eta}}{\psi_{\eta}}g^{2}+g_{\xi}\right)\\\
-\xi^{\eta}\left(g_{\eta}+2\frac{\psi_{\eta\eta}}{\psi_{\eta}}g\right)+(2k_{2}-k_{5})g,\end{array}$
(5.3)
$\begin{array}[]{c}\xi_{\eta}^{\eta}=-g\xi_{\eta}^{\xi}-\xi^{\xi}\left(g\frac{\psi_{\eta\eta}}{\psi_{\eta}}+g_{\eta}+\frac{S_{\xi}}{2(\gamma-1)S}\right)\\\
-\xi^{\eta}\left(\frac{\psi_{\eta\eta}}{\psi_{\eta}}+\frac{S_{\eta}}{2(\gamma-1)S}\right)+\frac{2(2\gamma-1)k_{2}-k_{5}}{2(\gamma-1)},\end{array}$
(5.4)

$\begin{array}[]{c}\xi_{\xi}^{\xi}=g\xi_{\eta}^{\xi}+\xi^{\xi}\left(g_{\eta}+\frac{\psi_{\eta\eta}}{\psi_{\eta}}g-\frac{S_{\xi}}{2(\gamma-1)S}\right)\\\
+\xi^{\eta}\left(\frac{\psi_{\eta\eta}}{\psi_{\eta}}-\frac{S_{\eta}}{2(\gamma-1)S}\right)-\frac{(2\gamma-3)k_{5}+2k_{2}}{2(\gamma-1)}.\end{array}$
(5.5)

where

$j_{1}=S_{\xi}-gS_{\eta},\,\,\,g=\frac{\psi_{\xi}}{\psi_{\eta}}.$ (5.6)

As a solution of equations (5.1)-(5.5) determines an admitted Lie group of
equations (3.1), they are called the defining equations.

The generators admitted for any functions $S$, $b_{01}$ and $b_{02}$, composes
a Lie algebra, called the kernel of admitted Lie algebras. A basis of this Lie
algebra consists of the generators

$\begin{array}[]{c}{\displaystyle
X_{1}=\frac{\partial}{\partial\varphi},\,\,\,X_{2}=\frac{\partial}{\partial\zeta},\,\,\,X_{3}=\frac{\partial}{\partial\chi},\,\,\,X_{4}=\frac{\partial}{\partial
t},}\\\ {\displaystyle
X_{5}=t\frac{\partial}{\partial\varphi},\,\,\,X_{6}=t\frac{\partial}{\partial\zeta},\,\,\,X_{7}=\zeta\frac{\partial}{\partial\varphi}-\varphi\frac{\partial}{\partial\zeta},\,\,\,X_{\lambda_{1}}=\lambda_{1}\frac{\partial}{\partial\chi},\,\,\,X_{\lambda_{2}}=t\lambda_{2}\frac{\partial}{\partial\chi}.}\end{array}$
(5.7)

The kernel extensions are discussed next.

###  5.1 Case $j_{1}\neq 0$

Introducing

$h_{1}=\xi^{\xi}\psi_{\xi}+\xi^{\eta}\psi_{\eta},\,\,\,h_{2}=\xi^{\xi}S_{\xi}+\xi^{\eta}S_{\eta},$

one finds

$\xi^{\xi}=(\psi_{\eta}j_{1})^{-1}(-S_{\eta}h_{1}+\psi_{\eta}h_{2}),\ \
\xi^{\eta}=(\psi_{\eta}j_{1})^{-1}(S_{\xi}h_{1}-\psi_{\eta}gh_{2}).$

From equation (5.4) one obtains

$h_{2}=S\left(2\frac{h_{1\eta}}{\psi_{\eta}}(1-\gamma)+2k_{2}(2\gamma-1)-k_{5}\right).$
(5.8)

Finding $h_{1\xi\eta}$ from equation (5.5), equation (5.3) becomes

$h_{1\xi}-h_{1\eta}g=0.$ (5.9)

Hence, $h_{1}=h_{1}(\psi)$, and equation (5.5) reduces to

$\begin{array}[]{c}{\displaystyle
h_{1\eta}j_{2}+h_{1}\left(\frac{(2\gamma+j_{2}-1)}{2(\gamma-1)}\frac{S_{\eta}}{S}-\frac{j_{1\eta}}{j_{1}}-\frac{\psi_{\eta\eta}}{\psi_{\eta}}\right)}\\\
{\displaystyle+\frac{\psi_{\eta}}{\gamma-1}\left(k_{2}j_{2}(1-2\gamma)+\frac{1}{2}k_{5}j_{2}-k_{2}+\frac{3-2\gamma}{2}k_{5}\right)=0,}\end{array}$
(5.10)

where

$j_{2}=j_{1}^{-2}\left(2(\gamma-1)S(j_{1\xi}-gj_{1\eta}+j_{1}g_{\eta})-(2\gamma-1)j_{1}^{2}\right).$
(5.11)

Notice that from the notation (5.11) one has

$j_{1\xi}=gj_{1\eta}-j_{1}g_{\eta}+\frac{j_{1}^{2}}{2(\gamma-1)S}(j_{2}+(2\gamma-1)).$
(5.12)

####  5.1.1 Case $j_{2}\neq 0$

From equation (5.10) one finds $h_{1\eta}$. Introducing the function

$j_{3}=j_{2\xi}-gj_{2\eta},$ (5.13)

the compatibility condition $(h_{1\xi})_{\eta}=(h_{1\eta})_{\xi}$ becomes

$h_{1}\mu+2\psi_{\eta}^{2}Sj_{1}j_{3}\tilde{k}_{2}=0,$ (5.14)

where ${\displaystyle\tilde{k}_{2}=k_{2}+\frac{(2\gamma-3)}{2}k_{5}}$, and

$\mu=j_{3}\left(2(\gamma-1)S(j_{1\eta}\psi_{\eta}+\psi_{\eta\eta}j_{1})-S_{\eta}\psi_{\eta}j_{1}(2\gamma-1)\right)-j_{2\eta}\psi_{\eta}j_{1}^{2}j_{2}.$
(5.15)

Let $j_{3}\mu\neq 0$. Introducing the function

$j_{4}=-2\mu^{-1}\psi_{\eta}^{2}Sj_{1}j_{3},$ (5.16)

equation (5.14) gives that $h_{1}=j_{4}\tilde{k}_{2}$. As for
$\tilde{k}_{2}=0$ there is no an extension of the kernel of admitted Lie
algebras, and because $h_{1}=h_{1}(\psi)$, then

$j_{4}=j_{4}(\psi).$

From definition of $j_{4}$ one finds

$\psi_{\eta\eta}=\frac{\psi_{\eta}}{(\gamma-1)}\left(\frac{(2\gamma-1)S_{\eta}}{2S}-\frac{(\gamma-1)j_{1\eta}}{j_{1}}+\frac{j_{1}j_{2}j_{2\eta}}{2Sj_{3}}+\frac{\psi_{\eta}}{j_{4}}\right).$

The compatibility condition $(\psi_{\eta\eta})_{\xi}=(\psi_{\xi})_{\eta\eta}$
gives

$j_{1\eta}=\frac{j_{1}}{j_{3}^{2}}(j_{3\eta}j_{2\xi}-j_{3\xi}j_{2\eta})+\frac{j_{1}S_{\eta}}{S}+\frac{j_{1j_{2\eta}}}{j_{3}}\left(\frac{j_{1}(j_{2}+1)}{2(\gamma-1)S}-g_{\eta}\right).$
(5.17)

The relation $(j_{1\xi})_{\eta}=(j_{1\eta})_{\xi}$ provides the condition

$\begin{array}[]{c}{\displaystyle
j_{3}^{2}(j_{2\eta}g_{\xi\eta}-j_{2\xi}g_{\eta\eta})+j_{3}(j_{5\xi}j_{2\eta}-j_{5\eta}j_{2\xi})}\\\\[8.61108pt]
{\displaystyle+(j_{3}g_{\eta}+2j_{5})(j_{3\eta}j_{3}-j_{2\eta}j_{5})=0,}\end{array}$
(5.18)

where $j_{5}=j_{3\xi}-gj_{3\eta}$.

Substituting $h_{1}$ into (5.10) one derives

$k_{5}=\tilde{k}_{2}\frac{1}{4(\gamma-1)^{2}}\left(2(\gamma-1)\frac{j_{4\eta}}{\psi_{\eta}}-j_{4}M+2(2\gamma-1)\right),$
(5.19)

where

$M=\frac{S_{\eta}j_{3}-j_{2\eta}j_{1}}{\psi_{\eta}Sj_{3}}.$ (5.20)

Direct calculations show that $M$ satisfies the relation

$M_{\xi}-gM_{\eta}=0,$

which means that $M=M(\psi)$.

For the existence of an extension of the kernel of admitted Lie algebras one
needs to assume that $k_{8}/\tilde{k}_{2}$ is constant. Thus,

$2(\gamma-1)\frac{j_{4\eta}}{\psi_{\eta}}-j_{4}M=k,$ (5.21)

where $k$ is some constant.

Equation (5.1) becomes

$M_{\eta}=-\frac{g\psi_{\eta}M}{2(\gamma-1)}\left(M+\frac{k}{j_{4}}\right).$
(5.22)

The extension of the kernel of admitted Lie algebras is defined by the
generator

$\begin{array}[]{c}X_{9}^{(1)}=\frac{j_{4}}{\psi_{\eta}j_{3}}\left(-j_{2\eta}\frac{\partial}{\partial\xi}+j_{2\xi}\frac{\partial}{\partial\eta}\right)-\frac{k+2(2\gamma-1)}{4(\gamma-1)^{2}}t\frac{\partial}{\partial
t}\\\
+\frac{k(\gamma-2)-2\gamma}{4(\gamma-1)^{2}}\left(\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}\right).\end{array}$
(5.23)

Summarizing, one can state that if the functions $\psi(\xi,\eta)$ and
$S(\xi,\eta)$ satisfy the conditions (5.17), (5.21) and (5.22), where
$j_{i},\,\,\,(i=1,2,3,4)$ are defined by the formulas (5.6), (5.11), (5.13)
and (5.16), then the extension of the kernel of admitted Lie algebras is
defined by the generator (5.23). Here condition (5.18) guarantees the
existence of the functions $\psi(\xi,\eta)$, $S(\xi,\eta)$ satisfying
conditions (5.17) and (5.21).

Case $j_{i}\neq 0,\,\,\,(i=1,2,3)$ and $\mu=0$. Equation (5.14) provides that
$\tilde{k}_{2}=0$. From $\mu=0$ one finds that

$\psi_{\eta\eta}=-\frac{j_{1\eta}}{j_{1}}\psi_{\eta}+\frac{1}{2(\gamma-1)S}\left((2\gamma-1)S_{\eta}\psi_{\eta}+j_{2\eta}\psi_{\eta}j_{1}j_{2}j_{3}^{-1}\right).$
(5.24)

The compatibility relation $(\psi_{\eta\eta})_{\xi}=(\psi_{\xi})_{\eta\eta}$
is

$\begin{array}[]{c}{\displaystyle
j_{1\eta}=\frac{j_{1}}{j_{3}^{2}}\left(j_{2\eta}\left(-g_{\eta}j_{3}-j_{5}+\frac{j_{1}j_{3}(j_{2}-3)}{2(\gamma-1)S}\right)+j_{3}\left(\frac{\gamma+1}{\gamma-1}\frac{S_{\eta}}{S}j_{3}+j_{3\eta}\right)\right)}-2\frac{j_{1}}{\gamma-1}\psi_{\eta}M\end{array},$
(5.25)

where $j_{5}=j_{3\xi}-gj_{3\eta}$. The compatibility condition
$(j_{1\eta})_{\xi}=(j_{1\xi})_{\eta}$ also coincides with (5.18).

Equation (5.10) becomes

$\begin{array}[]{c}{\displaystyle
h_{1\eta}+\frac{M\psi_{\eta}}{2(\gamma-1)}h_{1}+2k_{5}(\gamma-1)\psi_{\eta}=0}.\end{array}$
(5.26)

Equation (5.25) provides that

$M_{\xi}-gM_{\eta}=0,$ (5.27)

which also means that $M=M(\psi)$. Equation (5.1) reduces to

$h_{1}\nu-4k_{5}(\gamma-1)^{2}M\psi_{\eta}=0,$

where $\nu=2(\gamma-1)M_{\eta}-M^{2}\psi_{\eta}$.

Assuming that $\nu\neq 0$, one obtains

$h_{1}=k_{5}\lambda,$

where

$\lambda=\frac{4(\gamma-1)^{2}M\psi_{\eta}}{\nu}.$

For the existence of the extension of the kernel of admitted Lie algebras it
is necessary that $\lambda$ is constant, say $\lambda=k$:

$h_{1}=kk_{5}.$

Substituting the latter into (5.26),

$M=-\frac{4(\gamma-1)^{2}}{k}$ (5.28)

or

$\frac{S_{\eta}j_{3}-j_{2\eta}j_{1}}{\psi_{\eta}Sj_{3}}=-\frac{4(\gamma-1)^{2}}{k},$
(5.29)

and the extension of the kernel of admitted Lie algebras is defined by the
generator

$X_{9}^{(2)}=\frac{k}{(\gamma-2)\psi_{\eta}j_{3}}\left(j_{2\eta}\frac{\partial}{\partial\xi}-j_{2\xi}\frac{\partial}{\partial\eta}\right)-\frac{1}{(\gamma-2)}t\frac{\partial}{\partial
t}+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}.$
(5.30)

Let $\nu=0$, then $k_{5}M=0$.

Consider $M=0$ or

$S_{\eta}j_{2\xi}-j_{2\eta}S_{\xi}=0.$

The latter means that $j_{2}=j_{2}(S)$. Integrating (5.26), one obtains

$h_{1}=-2k_{5}(\gamma-1)\psi+k_{12}.$ (5.31)

The extension of the kernel of admitted Lie algebras is defined by the
generators

$X_{9}^{(3)}=\frac{2(\gamma-1)\psi}{(\gamma-2)\psi_{\eta}j_{3}}\left(-j_{2\eta}\frac{\partial}{\partial\xi}+j_{2\xi}\frac{\partial}{\partial\eta}\right)-\frac{1}{\gamma-2}t\frac{\partial}{\partial
t}+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}.$
(5.32)
$X_{10}^{(3)}=\frac{1}{\psi_{\eta}j_{3}}\left(-j_{2\eta}\frac{\partial}{\partial\xi}+j_{2\xi}\frac{\partial}{\partial\eta}\right).$
(5.33)

If $M\neq 0$, then $k_{5}=0$,

$M=-\frac{2(\gamma-1)}{\psi+k},$

with some constant $k$, and equation (5.26) reduces to

$\begin{array}[]{c}{\displaystyle
h_{1\eta}+\frac{M\psi_{\eta}}{2(\gamma-1)}h_{1}=0}.\end{array}$ (5.34)

Hence, $h_{1}=k_{12}/M$ and the extension of the kernel of admitted Lie
algebras is defined by the generator

$X_{9}^{(4)}=\frac{1}{M\psi_{\eta}j_{3}}\left(-j_{2\eta}\frac{\partial}{\partial\xi}+j_{2\xi}\frac{\partial}{\partial\eta}\right).$
(5.35)

Case $j_{1}j_{2}\neq 0$ and $j_{3}=0$. The assumption $j_{3}=0$ gives that
$j_{2}=j_{2}(\psi)$, and equation (5.14) becomes $h_{1}j_{2\eta}=0$. If
$j_{2\eta}\neq 0$, then $h_{1}=0$ and equation (5.10) leads to the condition

$j_{2}\left(\tilde{k}_{2}(1-2\gamma)+2(\gamma-1)^{2}k_{5}\right)-\tilde{k}_{2}=0.$

As $j_{2\eta}\neq 0$, then the latter equation provides that $\tilde{k}_{2}=0$
and $k_{5}=0$. Hence, for $j_{2\eta}\neq 0$ there is no an extension of the
kernel of admitted Lie algebras. Thus, one should assume that $j_{2\eta}=0$,
which gives that $j_{2}=k$, where $k\neq 0$ is constant. Equation (5.10)
reduces to

$\begin{array}[]{c}{\displaystyle
h_{1\eta}-h_{1}\lambda\psi_{\eta}+\beta\psi_{\eta}=0},\end{array}$ (5.36)

where

$\lambda=\frac{1}{k}\left(\frac{\psi_{\eta\eta}}{\psi_{\eta}}+\frac{j_{1\eta}}{j_{1}}-\frac{(2\gamma+k-1)S_{\eta}}{2(\gamma-1)S}\right),\,\,\,\beta=2k_{5}(\gamma-1)-\tilde{k}_{2}\frac{(2\gamma-1)k+1}{k(\gamma-1)}.$

Finding $\psi_{\eta\eta}$ from the latter notation of $\lambda$, the condition
$(\psi_{\eta\eta})_{\xi}=(\psi_{\xi})_{\eta\eta}$ provides that
$\lambda=\lambda(\psi)$.

Equation (5.1) becomes

${\displaystyle h_{1}j_{5}-\lambda\beta=0},$ (5.37)

where

$j_{5}=\frac{\lambda_{\eta}}{\psi_{\eta}}+\lambda^{2}.$

As $\lambda=\lambda(\psi)$, then $j_{5}=j_{5}(\psi)$.

Consider $j_{5}\neq 0$. Substituting ${\displaystyle
h_{1}=\beta\frac{\lambda}{j_{5}}}$ into (5.36), one gets

$\beta(\lambda j_{5\eta}+2j_{5}(\lambda^{2}-j_{5})\psi_{\eta})=0.$

If $\lambda j_{5\eta}+2j_{5}(\lambda^{2}-j_{5})\psi_{\eta}\neq 0$, then
$\beta=0$ or

$k_{5}=\tilde{k}_{2}\frac{k(2\gamma-1)+1}{2k(\gamma-1)^{2}}.$

The extension of the kernel of admitted Lie algebras is defined by the
generator

$\begin{array}[]{c}X_{9}^{(5)}=\frac{4S(\gamma-1)^{2}}{j_{1}}\left(-\frac{\partial}{\partial\xi}+g\frac{\partial}{\partial\eta}\right)+(k(2\gamma-1)+1)t\frac{\partial}{\partial
t}\\\
+(k\gamma+2-\gamma)\left(\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}\right).\end{array}$
(5.38)

If $\lambda j_{5\eta}+2j_{5}(\lambda^{2}-j_{5})\psi_{\eta}=0$, then the
extension of the kernel of admitted Lie algebras is defined by the generator
$X_{9}^{(6)}=X_{9}^{(5)}$ and one more generator

$\begin{array}[]{c}X_{10}^{(6)}=\frac{2(\gamma-1)\lambda}{(\gamma-2)\psi_{\eta}j_{1}j_{5}}\left((S_{\eta}+2(\gamma-1)\lambda
S\psi_{\eta})\frac{\partial}{\partial\xi}-(S_{\xi}+2(\gamma-1)\lambda
S\psi_{\xi})\frac{\partial}{\partial\eta}\right)\\\
-\frac{1}{\gamma-2}t\frac{\partial}{\partial
t}+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}.\end{array}$
(5.39)

Considering $j_{5}=0$, one obtains that $\lambda\beta=0$.

If $\lambda\neq 0$, then ${\displaystyle\beta=0}$ and the extension of the
kernel of admitted Lie algebras is defined by the generator
$X_{9}^{(7)}=X_{9}^{(5)}$ and by one more generator

$X_{10}^{(7)}=\frac{h_{11}}{\psi_{\eta}j_{1}}\left(-(S_{\eta}+2(\gamma-1)\lambda
S\psi_{\eta})\frac{\partial}{\partial\xi}+(S_{\xi}+2(\gamma-1)\lambda
S\psi_{\xi})\frac{\partial}{\partial\eta}\right),$ (5.40)

where $h_{11}(\psi)$ is the general solution of equation (5.36):

${\displaystyle h_{11}^{\prime}=h_{11}\lambda.}$ (5.41)

If $\lambda=0$, then solving equation (5.36), one derives

${\displaystyle h_{1}=-\beta\psi+k_{20}},$ (5.42)

where $k_{20}$ is an arbitrary constant. The extension of the kernel of
admitted Lie algebras is defined by the generators

$\begin{array}[]{c}X_{9}^{(8)}={\displaystyle\frac{1}{(\gamma-1)k\psi_{\eta}j_{1}}\left(-((k(2\gamma-1)+1)S_{\eta}\psi+2(\gamma-1)S\psi_{\eta})\frac{\partial}{\partial\xi}\right.}\\\
{\displaystyle\left.+((k(2\gamma-1)+1)S_{\xi}\psi+2(\gamma-1)S\psi_{\xi})\frac{\partial}{\partial\eta}\right)}\\\
{\displaystyle+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi},}\end{array}$
(5.43)
$X_{10}^{(8)}=\frac{2(\gamma-1)\psi}{(\gamma-2)\psi_{\eta}j_{1}}\left(-S_{\eta}\frac{\partial}{\partial\xi}+S_{\xi}\frac{\partial}{\partial\eta}\right),$
(5.44)

$X_{11}^{(8)}=\frac{1}{\psi_{\eta}j_{1}}\left(-S_{\eta}\frac{\partial}{\partial\xi}+S_{\xi}\frac{\partial}{\partial\eta}\right),$

####  5.1.2 Case $j_{1}\neq 0$ and $j_{2}=0$.

Equation (5.10) becomes

${\displaystyle h_{1}N-\frac{1}{\gamma-1}(k_{2}+k_{5}(2\gamma-3))=0,}$ (5.45)

where

$N=\frac{1}{\psi_{\eta}}\left(\frac{(2\gamma-1)S_{\eta}}{2(\gamma-1)S}-\frac{j_{1\eta}}{j_{1}}-\frac{\psi_{\eta\eta}}{\psi_{\eta}}\right).$

Conditions (5.12) provide that $N=N(\psi)$.

Assume that $N=0$. Finding $\psi_{\eta\eta}$ from the condition $N=0$:

$\psi_{\eta\eta}=\psi_{\eta}\left(\frac{(2\gamma-1)S_{\eta}}{2(\gamma-1)S}-\frac{j_{1\eta}}{j_{1}}\right),$
(5.46)

one checks that $(\psi_{\xi})_{\eta\eta}=(\psi_{\eta\eta})_{\xi}$. Equation
(5.45) reduces to the equation

$k_{2}=k_{5}\left(\frac{3}{2}-\gamma\right),$

and equation (5.1) becomes

$\left(\frac{h_{1\eta}}{\psi_{\eta}}\right)_{\eta}=0.$
$k_{\eta}=k_{1}\psi_{\eta}$

As $h_{1}=h_{1}(\psi)$, one finds that

$h_{1}=k_{21}\psi+k_{20,}$

where $k_{21}$ and $k_{20}$ are arbitrary constants. The extension of the
kernel of admitted Lie algebras (5.7) is defined by the generators

$X_{9}^{(9)}=\frac{4(\gamma-1)^{2}S}{(\gamma-2)j_{1}}\left(\frac{\partial}{\partial\xi}-g\frac{\partial}{\partial\eta}\right)+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}-\frac{t}{(\gamma-2)}\frac{\partial}{\partial
t},$ (5.47)

$X_{10}^{(9)}=\frac{1}{\psi_{\eta}j_{1}}\left(-(\psi
S_{\eta}+2(\gamma-1)S\psi_{\eta})\frac{\partial}{\partial\xi}+(\psi
S_{\xi}+2(\gamma-1)S\psi_{\xi})\frac{\partial}{\partial\eta}\right),$ (5.48)

$X_{11}^{(9)}=\frac{1}{\psi_{\eta}j_{1}}\left(-S_{\eta}\frac{\partial}{\partial\xi}+S_{\xi}\frac{\partial}{\partial\eta}\right)$
(5.49)

Assuming that $N\neq 0$, one can introduce the function $P(\psi)$ instead of
the function $N(\psi)$ by the formula

$P=\frac{1}{(\gamma-1)N}$

or the function $P$ is introduced by the formula

$\psi_{\eta\eta}=\psi_{\eta}\left(\frac{(2\gamma-1)S_{\eta}}{2(\gamma-1)S}-\frac{j_{1\eta}}{j_{1}}-\frac{\psi_{\eta}}{(\gamma-1)P}\right).$
(5.50)

As in the previous case the compatibility condition
$(\psi_{\xi})_{\eta\eta}=(\psi_{\eta\eta})_{\xi}$ is also satisfied. Equation
(5.45) gives

$h_{1}=\frac{2k_{2}+k_{5}(2\gamma-3)}{2}P.$

As $P=P(\psi)$, then
$\left(\frac{P_{\eta}}{\psi_{\eta}}\right)_{\eta}=P^{\prime\prime}\psi_{\eta}$,
and equation (5.1) reduces to

$P^{\prime\prime}(2k_{2}+k_{5}(2\gamma-3))=0.$

If $P^{\prime\prime}\neq 0$, then the extension of the kernel of admitted Lie
algebras (5.7) is defined by the generator $X_{9}^{(9)}$

$X_{9}^{(10)}=\frac{4(\gamma-1)^{2}S}{(\gamma-2)j_{1}}\left(\frac{\partial}{\partial\xi}-g\frac{\partial}{\partial\eta}\right)+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}-\frac{t}{(\gamma-2)}\frac{\partial}{\partial
t},$ (5.51)

and if $P^{\prime\prime}=0$, then there is one more admitted generator

$\begin{array}[]{c}X_{10}^{(10)}={\displaystyle\frac{1}{\psi_{\eta}j_{1}}\left(-P\left(S_{\eta}\frac{\partial}{\partial\xi}-S_{\xi}\frac{\partial}{\partial\eta}\right)+2S((\gamma-1)P^{\prime}-(2\gamma-1)))\left(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta}\right)\right)}\\\
{\displaystyle+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}.}\end{array}$
(5.52)

###  5.2 Case $j_{1}=0$

The condition $j_{1}=0$ provides that $S=S(\psi)$, equation (5.2) is
satisfied, and equation (5.1) becomes

$(\xi^{\eta}+\xi^{\xi}g)S_{\eta}g_{1}+S(2k_{2}(1-2\gamma)+k_{5})=0,$ (5.53)

where

$g_{1}=2(\gamma-1)S\frac{\psi_{\eta\eta}S_{\eta}-S_{\eta\eta}\psi_{\eta}}{\psi_{\eta}^{3}}+(2\gamma-1)\frac{S_{\eta}^{2}}{\psi_{\eta}^{2}}$

Assume that $g_{1}\neq 0$. From the latter equation one finds

$\xi^{\eta}=-\xi^{\xi}g-\frac{S}{g_{1}S_{\eta}}(2(2\gamma-1)k_{2}-k_{5}).$

Substituting $\xi^{\eta}$ into (5.3) and (5.4), they reduce to the single
equation

$g_{1\eta}(2(2\gamma-1)k_{2}-k_{5})=0.$

Let $g_{1\eta}\neq 0$, then $k_{5}=2(2\gamma-1)k_{2}$, and equation (5.5)
reduces to the quasilinear first-order partial differential equation for the
single function $\xi^{\xi}$:

$\xi_{\xi}^{\xi}-g\xi_{\eta}^{\xi}=g_{\eta}\xi^{\xi}+4(\gamma-1)k_{2}.$

The general solution of the latter equation can be written as follows

$\xi^{\xi}=\psi_{\eta}\left(h_{11}-4(\gamma-1)k_{2}h_{12}\right),$

where $h_{11}=h_{11}(\psi)$ is an arbitrary function and $h_{12}(\xi,\eta)$ is
an arbitrary solution of the linear equation

$h_{12\xi}-gh_{12\eta}+\psi_{\eta}^{-1}=0.$

The extension of the kernel of admitted Lie algebras (5.7) is defined by the
generators

$X_{9}^{(12)}=h_{11}\psi_{\eta}\left(\frac{\partial}{\partial\xi}-g\frac{\partial}{\partial\eta}\right),$
(5.54)

$X_{10}^{(12)}=\begin{array}[]{c}{\displaystyle\frac{2(\gamma-1)}{\gamma}h_{12}\left(-\psi_{\eta}\frac{\partial}{\partial\xi}+\psi_{\xi}\frac{\partial}{\partial\eta}\right)+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}+\frac{2\gamma-1}{\gamma}t\frac{\partial}{\partial
t}.}\end{array}$ (5.55)

Let $g_{1\eta}=0$, say $g_{1}=k$, where $k\neq 0$ is constant. Equation (5.5)
becomes

$\begin{array}[]{c}\xi_{\xi}^{\xi}-g\xi_{\eta}^{\xi}=g_{\eta}\xi^{\xi}+\frac{S\psi_{\eta\eta}}{kS_{\eta}\psi_{\eta}}\left(2k_{2}(2\gamma-1)-k_{5}\right)\\\
+\frac{1}{2k(\gamma-1)}(2k_{2}(k+1-2\gamma)+k_{5}(k(2\gamma-3)+1).\end{array}$

The general solution of the latter equation is written in the form

$\xi^{\xi}=\psi_{\eta}(h_{11}+k_{2}h_{12}+k_{5}h_{13}),$

where $h_{11}=h_{11}(\psi)$ is an arbitrary function, $h_{12}(\xi,\eta)$ and
$h_{13}(\xi,\eta)$ are arbitrary solutions of the linear equations

$h_{12\xi}-gh_{12\eta}=\frac{(1-\gamma)(2\gamma-1)\psi_{\eta\eta}S}{kS_{\eta}\psi_{\eta}^{2}}+\frac{2\gamma-1-k}{2k\psi_{\eta}},$
$h_{13\xi}-gh_{13\eta}=\frac{(1-\gamma)\psi_{\eta\eta}S}{kS_{\eta}\psi_{\eta}^{2}}+\frac{k(2\gamma-3)+1}{2k\psi_{\eta}}.$

The extension of the kernel of admitted Lie algebras (5.7) is defined by the
generators $X_{9}^{(13)}=X_{9}^{(12)}$ and

$X_{10}^{(13)}=\begin{array}[]{c}\frac{2h_{12}}{\gamma-1}\left(-\psi_{\eta}\frac{\partial}{\partial\xi}+\psi_{\xi}\frac{\partial}{\partial\eta}\right)+\frac{2(2\gamma-1)S}{kS_{\eta}}\frac{\partial}{\partial\eta}+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}.\end{array}$

$X_{11}^{(13)}=\begin{array}[]{c}\frac{2h_{13}}{\gamma-1}\left(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta}\right)-\frac{2S}{kS_{\eta}}\frac{\partial}{\partial\eta}+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}+2t\frac{\partial}{\partial
t}.\end{array}$

Case $g_{1}=0$. Equation (5.53) gives that $k_{5}=2k_{2}(2\gamma-1)$, and
equation (5.5) takes the form

$\xi_{\xi}^{\xi}-g\xi_{\eta}^{\xi}=(g_{\eta}+g\frac{g_{2\eta}}{g_{2}})\xi^{\xi}+\frac{g_{2\eta}}{g_{2}}\xi^{\eta}+4k_{2}(\gamma-1),$
(5.56)

where

$\frac{g_{2\eta}}{g_{2}}=\frac{\psi_{\eta\eta}}{\psi_{\eta}}-\frac{S_{\eta}}{2(\gamma-1)S},$
(5.57)

where such denotation is introduced for further simplifications.

Case $g_{2\eta}\neq 0$. Finding $\xi^{\eta}$ from equation (5.56), and
substituting it into equations (5.3) and (5.4), one obtains two second-order
equations for $\xi^{\xi}$. These equations can be simplified by the
substitution

$\xi_{\xi}^{\xi}=g\xi_{\eta}^{\xi}+g_{\eta}\xi^{\xi}+4(\gamma-1)k_{2}+\frac{g_{2\eta}}{g_{2}^{2}}S^{-1/(\gamma-1)}\left(k_{2}+h\right),$

where $h(\xi,\eta)$ is some unknown function. Equations (5.3) and (5.4)
become, respectively,

$h_{\xi}=0,\,\,\,h_{\eta}=0.$ (5.58)

Hence, $h$ is constant, say $h=k_{20}$.

The general solution of the latter equation is presented in the form

$\xi^{\xi}=\psi_{\eta}(h_{11}+4(\gamma-1)k_{2}h_{12}+h_{13}k_{20}),$

where $h_{11}=h_{11}(\psi)$ is arbitrary function, and $h_{12}(\xi,\eta)$ and
$h_{13}(\xi,\eta)$ are arbitrary solutions of the linear equations

$\begin{array}[]{c}h_{12\xi}-gh_{12\eta}=\psi_{\eta}^{-1}\left(\frac{g_{2\eta}}{4(\gamma-1)g_{2}^{2}}S^{-1/(\gamma-1)}+1\right).\\\
h_{13\xi}-gh_{12\eta}=S^{-1/(\gamma-1)}\frac{g_{2\eta}}{\psi_{\eta}g_{2}^{2}}.\end{array}$
(5.59)

The extension of the kernel of admitted Lie algebras (5.7) is defined by the
generators (5.54) and (5.55):

$X_{9}^{(14)}=h_{11}\left(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta}\right),$

$X_{10}^{(14)}=\begin{array}[]{c}{\displaystyle\frac{2(\gamma-1)}{\gamma}h_{12}\left(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta}\right)+\frac{1}{2\gamma
g_{2}}S^{-1/(\gamma-1)}\frac{\partial}{\partial\eta}+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}+\frac{2\gamma-1}{\gamma}t\frac{\partial}{\partial
t},}\end{array}$
$X_{11}^{(14)}=h_{13}\left(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta}\right)+\frac{1}{g_{2}}S^{-1/(\gamma-1)}\frac{\partial}{\partial\eta}.$

Let $g_{2\eta}=0$.

$\frac{\psi_{\eta\eta}}{\psi_{\eta}}-\frac{S_{\eta}}{2(\gamma-1)S}=0\Rightarrow\left(\psi_{\eta}S^{-1/(2(\gamma-1))}\right)_{\eta}=0.$

The compatibility condition $(\psi_{\eta\eta})_{\xi}=(\psi_{\xi})_{\eta\eta}$
gives that $g(\xi,\eta)$ is a linear function with respect to $\eta$, say

$g=-\frac{\mu_{1}^{\prime\prime}}{\mu_{1}^{\prime}}\eta+\mu_{2}^{\prime}\mu_{1}^{\prime},$

where $\mu_{1}(\xi)$ and $\mu_{2}(\xi)$ are some functions such that
$\mu_{1}^{\prime}\neq 0$. Here the representation for $g$ is chosen for
convenience of further integration. In particular, solving the equation
$\psi_{\xi}=g\psi_{\eta}$, one finds

$\psi=\psi(z),\,\,\,z=\frac{\eta}{\mu_{1}^{\prime}}+\mu_{2}.$

The relation $g_{2}=0$ provides that

$\psi^{\prime}=qS^{\frac{1}{2(\gamma-1)}},$

where $q$ is constant. Notice that if one considers $S=S(z)$, then the
relation $g_{2\eta}=0$ gives

$S^{\prime}=\frac{2(\gamma-1)S\psi^{\prime\prime}}{\psi^{\prime}\,{}^{2}}.$

The condition $g_{1}=0$ leads to the relation

$\psi^{\prime}\psi^{\prime\prime\prime}-3\psi^{\prime\prime}{}^{2}=0.$

Introducing $h_{1}=\xi^{\eta}+g\xi^{\xi}$, one derives

$\xi^{\eta}=h_{1}-g\xi^{\xi}.$

Then equation (5.3) reduces to

$\left(h_{1}S^{1/(\gamma-1)}\right)_{\eta}=0,$

which gives

$h_{1}=\mu_{3}S^{-1/(\gamma-1)},$

where $\mu_{3}(\xi)$ is an arbitrary function. Substituting $h_{1}$ into
equation (5.4), one obtains that $\mu_{3}=k_{20}\mu_{1}^{\prime}$ with
constant $k_{20}$. Equation (5.5) takes the form

$\xi_{\xi}^{\xi}+\left(\frac{\mu_{1}^{\prime\prime}}{\mu_{1}^{\prime}}\eta-\mu_{2}^{\prime}\mu_{1}^{\prime}\right)\xi_{\eta}^{\xi}=-\frac{\mu_{1}^{\prime\prime}}{\mu_{1}^{\prime}}\xi^{\xi}+4(\gamma-1)k_{2}.$

The general solution of the latter equation is

$\xi^{\xi}=k_{2}\frac{4(\gamma-1)\mu_{1}}{\mu_{1}^{\prime}}+\frac{1}{\mu_{1}^{\prime}}F(z),$

where $F(z)$ is an arbitrary function.

The extension of the kernel of admitted Lie algebras (5.7) is defined by the
generators

$X_{9}^{(16)}=F\left(z_{\eta}\frac{\partial}{\partial\xi}-z_{\xi}\frac{\partial}{\partial\eta}\right),$

${\displaystyle\begin{array}[]{c}{\displaystyle
X_{10}^{(16)}=\frac{2(\gamma-1)}{\gamma}\mu_{1}\left(z_{\eta}\frac{\partial}{\partial\xi}-z_{\xi}\frac{\partial}{\partial\eta}\right)+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}+\frac{2\gamma-1}{\gamma}t\frac{\partial}{\partial
t},}\end{array}}$

$X_{11}^{(16)}=\mu_{1}^{\prime}S^{-1/(\gamma-1)}\frac{\partial}{\partial\eta},$

##  6 Nonisentropic case with $b_{01}^{2}+b_{02}^{2}=0$

In this case $H_{1}=0$ and $H_{2}=0$, and equation (3.1b) is integrated

$\chi=t\chi_{1}+\chi_{0},$

where $\chi_{0}(\xi,\eta)$ and $\chi_{1}(\xi,\eta)$ are arbitrary functions.
Then the variable $\chi(t,\xi,\eta)$ is excluded from the consideration. It is
also assumed that $S_{\eta}\neq 0$. Partially solving the determining
equations, one derives that $\xi^{\xi}=\xi^{\xi}(\xi,\eta)$,
$\xi^{\eta}=\xi^{\eta}(\xi,\eta)$, and

$\zeta^{\varphi}=k_{1}\zeta+(k_{2}+\frac{1}{2}k_{5})\varphi+tk_{7}+k_{8},$
$\zeta^{\zeta}=(k_{2}+\frac{1}{2}k_{5})\zeta-k_{1}\varphi+tk_{9}+k_{10},$
$\zeta^{\chi}=\chi\zeta^{1}+t\zeta^{2}+\zeta^{3},$ $\xi^{t}=k_{5}t+k_{6},$

where $\zeta^{i}(\xi,\eta),\,\,\,(i=1,2,3,4)$ are arbitrary functions. The
kernel of admitted Lie algebras (5.7) in these cases is extended by the
generators of the form

$Y=(\chi\zeta^{1}+t\zeta^{2}+\zeta^{3})\frac{\partial}{\partial\chi}.$

The remaining equations are

$S(\xi^{\eta}S_{\eta}+\xi^{\xi}S_{\xi})_{\xi}-S_{\xi}(\xi^{\eta}S_{\eta}+\xi^{\xi}S_{\xi})=0,$
(6.1) $\begin{array}[]{c}\begin{array}[]{c}{\displaystyle
S(\xi_{\eta}^{\xi}S_{\xi}-\xi_{\xi}^{\xi}S_{\eta})+\xi^{\xi}\left(SS_{\xi\eta}-\frac{\gamma}{\gamma-1}S_{\xi}S_{\eta}\right)+\xi^{\eta}\left(SS_{\eta\eta}-\frac{\gamma}{\gamma-1}S_{\eta}^{2}\right)}\\\
+SS_{\eta}\left(2k_{2}+\frac{\gamma}{\gamma-1}k_{5}\right)=0,\end{array}\end{array}$
(6.2)

$\begin{array}[]{c}\xi^{\eta}\left((\gamma-1)b_{03\eta}-\frac{S_{\eta}}{2S}b_{03}\right)+\xi^{\xi}\left((\gamma-1)b_{03\xi}-\frac{S_{\xi}}{2S}b_{03}\right)\\\
-b_{03}(k_{2}(\gamma-2)+k_{5}(1-\frac{\gamma}{2}))=0,\end{array}$ (6.3)

$\xi_{\eta}^{\eta}+\xi_{\xi}^{\xi}+\frac{1}{(\gamma-1)S}(S_{\eta}\xi^{\eta}+S_{\xi}\xi^{\xi})=\frac{1}{\gamma-1}(2\gamma
k_{2}+(\gamma-2)k_{5}).$ (6.4)

Notice that if $S$ is not constant, then the latter equation can be reduced to
the form

$S(\xi^{\eta}S_{\eta}+\xi^{\xi}S_{\xi})_{\eta}-S_{\eta}(\xi^{\eta}S_{\eta}+\xi^{\xi}S_{\xi})=0.$

Let $h_{1}=\xi^{\xi}S_{\xi}+\xi^{\eta}S_{\eta}$,
$h_{2}=\xi^{\xi}b_{03\xi}+\xi^{\eta}b_{03\eta}$, and
$f_{2}=b_{03\xi}S_{\eta}-b_{03\eta}S_{\xi}.$

###  6.1 Case $f_{2}\neq 0$

One can derive

$\xi^{\xi}=f_{2}^{-1}(h_{2}S_{\eta}-h_{1}b_{03\eta}),\,\,\,\xi^{\eta}=f_{2}^{-1}(-h_{2}S_{\xi}+h_{1}b_{03\xi}),\,\,$

From equations (6.1) and (6.4) one finds

$h_{1}=Sk_{20},$

where $k_{20}$ is constant. Equation (6.3) gives

$h_{2}=b_{03}\frac{(\gamma-2)(2k_{2}-k_{5})+k_{20}}{2(\gamma-1)}.$

Let $k_{2}=\frac{(\gamma-2)k_{5}+\tilde{k}_{2}}{2(\gamma-2)}$. Equation (6.2)
becomes

$k_{2}=f_{3}\tilde{k}_{2}+bk_{20}$ (6.5)

where

$f_{3}=\frac{1}{4(\gamma-1)}\left(\frac{b_{03}}{f_{2}^{2}}(S_{\xi}f_{2\eta}-S_{\eta}f_{2\xi})-\frac{\gamma+2}{\gamma-2}\right),\,\,\,b=f_{4}+f_{3}+\frac{\gamma}{2(\gamma-2)},$

and

$f_{4}=\frac{S}{S_{\eta}}\left(\frac{b_{03\eta}}{b_{03}}\left(2(1-\gamma)f_{3}-\frac{\gamma+2}{2(\gamma-2)}\right)-\frac{f_{2\eta}}{2f_{2}}\right).$

Differentiating $k_{2}$ with respect to $\xi$ and $\eta$, one derives that it
is necessary to study the cases (a) $f_{3}\neq const$ and (b) $f_{3}=const$.

If $f_{3}\neq const$, then one can assume that $f_{3}\neq 0$. Hence, for
existence of an extension of the kernel of admitted Lie algebras one obtains
from equation (6.5) that there exist constants $k$ and $q$ such that
$b=kf_{3}+q$ and $\tilde{k}_{2}=-kk_{20}$.

Thus, the extension of the kernel of admitted Lie algebras is defined by the
generator

$\begin{array}[]{c}X_{9}^{(17)}={\displaystyle\left(q-\frac{k}{2(\gamma-2)}\right)(\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta})+qt\frac{\partial}{\partial
t}}\\\
{\displaystyle+\frac{1}{2f_{2}}\left(-\left(2Sb_{03\eta}+(k-1)b_{03}S_{\eta}\right)\frac{\partial}{\partial\xi}+\left(2Sb_{03\xi}+(k-1)b_{03}S_{\xi}\right)\frac{\partial}{\partial\eta}\right).}\end{array}$

Notice that as $S_{\eta}\neq 0$, then from the definition of $f_{3}$ one can
find $f_{2\xi}$. Finding $f_{2\eta}$ from the equation $b=kf_{3}+q$, the
compatibility condition $(f_{2\xi})_{\eta}-(f_{2\eta})_{\xi}=0$ gives

$2(\gamma-1)S(b_{03\eta}f_{3\xi}-b_{03\xi}f_{3\eta})+(k-1)b_{03}(S_{\eta}f_{3\xi}-S_{\xi}f_{3\eta})=0.$

Consider case $f_{3}=const$.

Assume that $f_{4}$ is constant, say $f_{4}=m$. In this case
$f_{2}=qb_{03}^{q_{2}}S^{q_{3}}$, where $q$ is an arbitrary constant and

$q_{2}=-4(\gamma-1)f_{3}-\frac{\gamma+2}{\gamma-2},\,\,\,q_{3}=-2m.$

Thus,

$b_{03\xi}=(b_{03\eta}S_{\xi}+f_{2})/S_{\eta},$

and the extension of admitted Lie algebras occurs by the generators

$\begin{array}[]{c}X_{9}^{(18)}={\displaystyle\left(f_{3}+m+\frac{\gamma}{2(\gamma-2)}\right)\left(\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+t\frac{\partial}{\partial
t}\right)}\\\
{\displaystyle+\frac{1}{2qb_{03}^{q_{2}}S^{q_{3}}}\left(\left(-2Sb_{03\eta}+\frac{b_{03}S_{\eta}}{\gamma-1}\right)\frac{\partial}{\partial\xi}-\left(-2Sb_{03\xi}+\frac{b_{03}S_{\xi}}{\gamma-1}\right)\frac{\partial}{\partial\eta}\right),}\end{array}$
$\begin{array}[]{c}X_{10}^{(18)}={\displaystyle\left(f_{3}+\frac{1}{2(\gamma-2)}\right)\left(\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}\right)+f_{3}t\frac{\partial}{\partial
t}}\\\
{\displaystyle+\frac{S^{-q_{3}}b_{03}^{1-q_{2}}}{2q(\gamma-1)}\left(S_{\eta}\frac{\partial}{\partial\xi}-S_{\xi}\frac{\partial}{\partial\eta}\right).}\end{array}$

If $f_{4}$ is not constant, then $k_{20}=0$, $f_{4}=f_{4}(S)$, and

$b_{03\xi}=(b_{03\eta}S_{\xi}+f_{2})/S_{\eta}.$

The extension of the kernel of admitted Lie algebras consists of the generator

$X_{9}^{(19)}=\left(f_{3}+\frac{1}{2(\gamma-2)}\right)\left(\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}\right)+f_{3}t\frac{\partial}{\partial
t}+\frac{b_{03}}{2(\gamma-1)f_{2}}\left(S_{\eta}\frac{\partial}{\partial\xi}-S_{\xi}\frac{\partial}{\partial\eta}\right).$

###  6.2 Case $f_{2}=0$

In this case $b_{03}=b_{03}(S)$. From equations (6.1) and (6.4) one finds that
$h_{1}=Sk_{20}$. Equation (6.3) becomes k

$k_{5}=2\tilde{k}_{2}+f_{5}k_{20},$

where

$f_{5}=\frac{2(1-\gamma)}{(\gamma-2)}\frac{Sb_{03\eta}}{S_{\eta}b_{03}}+\frac{1}{\gamma-2}.$

Notice that $f_{5}=f_{5}(S)$.

If $f_{5}\neq const$, then $k_{20}=0$, the general solution of equation (6.2)
can be presented in the form

$\xi^{\xi}=S_{\eta}(\psi_{1}+\psi_{2}k_{2}).$

Substituting the latter into (6.2), one finds that $\psi_{1}(S)$ is an
arbitrary function and the function $\psi_{2}$ is a solution of the equation

$\psi_{2\xi}S_{\eta}-\psi_{2\eta}S_{\xi}=4.$

The extension of the kernel of admitted Lie algebras occurs by the generators

$X_{9}^{(20)}=\psi_{1}\left(S_{\eta}\frac{\partial}{\partial\xi}-S_{\xi}\frac{\partial}{\partial\eta}\right),$
$X_{10}^{(20)}=2\left(\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+t\frac{\partial}{\partial
t}\right)+\psi_{2}\left(S_{\eta}\frac{\partial}{\partial\xi}-S_{\xi}\frac{\partial}{\partial\eta}\right).$

If $f_{5}=k$, then the extension of the kernel of admitted Lie algebras is
defined by the generators $X_{9}^{21}=X_{9}^{20}$ and
$X_{10}^{21}=X_{10}^{20}$ and by one more generator

$X_{11}^{(21)}=k\left(\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+2t\frac{\partial}{\partial
t}\right)+2\psi_{3}\left(S_{\eta}\frac{\partial}{\partial\xi}-S_{\xi}\frac{\partial}{\partial\eta}\right)+2\frac{S}{S_{\eta}}\frac{\partial}{\partial\eta},$

where the function $\psi_{3}$ is a solution of the equation

$S_{\eta}\psi_{3\xi}-S_{\xi}\psi_{3\eta}=\frac{SS_{\eta\eta}}{S_{\eta}^{2}}+\frac{k(\gamma-2)-\gamma}{\gamma-1}.$

##  7 Isentropic case with $b_{01}^{2}+b_{02}^{2}\neq 0$

For the isentropic case, equations (5.1)-(5.5) reduce to the following

$\xi_{\xi}^{\eta}=-\xi_{\eta}^{\xi}g^{2}+\xi^{\eta}(-g_{\eta}-2\psi_{\eta\eta}\psi_{\eta}^{-1}g)+\xi^{\xi}(-2g_{\eta}g-g_{\xi}-2\psi_{\eta\eta}\psi_{\eta}^{-1}g^{2})+g(2k_{2}-k_{5}),$
(7.1)
$eq_{1}+eq_{3}=\begin{array}[]{c}\xi_{\xi}^{\xi}+\xi_{\eta}^{\eta}=2\frac{2\gamma
k_{2}+(\gamma-2)k_{5}}{\gamma-1},\end{array}$ (7.2)

$\xi_{\xi}^{\xi}=\xi_{\eta}^{\xi}g+\xi^{\eta}\psi_{\eta\eta}\psi_{\eta}^{-1}+\xi^{\xi}(g_{\eta}+\psi_{\eta\eta}\psi_{\eta}^{-1}g)+\frac{2\gamma
k_{5}+2k_{2}-3k_{5}}{\gamma-1},$ (7.3)

Assume that $\psi_{\eta\eta}\neq 0$. Substituting $\xi^{\eta}$, found from
equation (7.3), into (7.1) and (7.2), one can integrate them

$\xi_{\xi}^{\xi}-\xi_{\eta}^{\xi}g-\xi^{\xi}g_{\eta}+\frac{\psi_{\eta\eta}}{\psi_{\eta}^{2}}\left(\frac{((1-2\gamma)k_{2}+\frac{1}{2}k_{5})\psi}{\gamma-1}+k_{20}\right)-\frac{k_{2}+(\gamma-\frac{3}{2})k_{5}}{\gamma-1}=0,$
(7.4)

The latter is a linear equation for the function $\xi^{\xi}$. The general
solution of this equation can be found in the form

$\xi^{\xi}=\psi_{\eta}(\psi_{1}+\psi_{2}k_{2}+\psi_{3}k_{20}+\psi_{4}k_{5}),$

where $\psi_{i}=\psi_{i}(\xi,\eta),\ (i=1,2,3,4)$ are some functions.
Substituting $\xi^{\xi}$ into (7.4) and splitting it, one obtains

$\psi_{1\xi}=\psi_{1\eta}g,$
$\psi_{2\xi}=\psi_{2\eta}g+\frac{1}{\gamma-1}\left(\psi_{\eta\eta}\psi_{\eta}^{-3}\psi(2\gamma-1)+\psi_{\eta}^{-1}\right),$
$\psi_{3\xi}=-\psi_{\eta\eta}\psi_{\eta}^{-3}+\psi_{3\eta}g,$
$\psi_{4\xi}=\psi_{4\eta}g+\frac{1}{2(\gamma-1}\left(\psi_{\eta\eta}\psi_{\eta}^{-3}\psi+\psi_{\eta}^{-1}(2\gamma-3)\right).$

The extension of admitted Lie algebras is defined by the generators

$X_{9}^{(22)}=\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}+2t\frac{\partial}{\partial
t}+2\psi_{4}(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta})-\frac{\psi}{(\gamma-1)\psi_{\eta}}\frac{\partial}{\partial\eta}$
$X_{10}^{(22)}=\psi_{3}(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta})-\psi_{\eta}^{-1}\frac{\partial}{\partial\eta},$
$X_{11}^{(22)}=\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}+\psi_{2}(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta})+\frac{(2\gamma-1)\psi}{(\gamma-1)\psi_{\eta}}\frac{\partial}{\partial\eta},$
$X_{12}^{(22)}=\psi_{1}(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta}).$

$\psi=\frac{\eta}{\mu_{1}^{\prime}}+\mu_{2}$

equation (7.3) can be integrated

$\xi^{\xi}=\frac{1}{(\gamma-1)\mu_{1}^{\prime}}((\gamma-1)h+(\gamma-\frac{3}{2})k_{5}\mu_{1}+k_{2}\mu_{1}),$

where $h(\psi)$ is an arbitrary function. Integrating equation (7.2), one
finds

$\xi^{\eta}+\psi_{\xi}h-\alpha\eta=f\mu_{1}^{\prime},$

where

$\alpha=\mu_{1}^{\prime\prime}\frac{\mu_{1}(k_{2}+(\gamma-\frac{3}{2})k_{5})}{(\gamma-1)\mu_{1}^{\prime}{}^{2}}+\frac{(2\gamma-1)k_{2}-\frac{1}{2}k_{5}}{\gamma-1},$

and the function $f(\xi)$ is a function of the integration. Substituting
$\xi^{\eta}$ into equation (7.1), one finds the function $f$:

$f=\alpha_{1}k_{2}+\alpha_{2}k_{5}+k_{20},$

where

$\alpha_{1}=-\frac{\mu_{2}^{\prime}\mu_{1}}{(\gamma-1)\mu_{1}^{\prime}}+\frac{2\gamma-1}{\gamma-1}\mu_{2},$
$\alpha_{2}=-\frac{(\gamma-3/2)\mu_{2}^{\prime}\mu_{1}}{(\gamma-1)\mu_{1}^{\prime}}-\frac{\mu_{2}}{2(\gamma-1)},$

and $k_{20}$ is a constant of integration.

The extension of admitted Lie algebras is defined by the generators

$X_{11}^{(23)}=h\left(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta}\right),$
$X_{9}^{(23)}=\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}+2t\frac{\partial}{\partial
t}+\frac{(2\gamma-3)\mu_{1}}{\gamma-1}(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta})-\frac{\psi\mu_{1}^{\prime}}{\gamma-1}\frac{\partial}{\partial\eta},$

$X_{10}^{(23)}=\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\chi\frac{\partial}{\partial\chi}+\frac{\mu_{1}}{\gamma-1}(\psi_{\eta}\frac{\partial}{\partial\xi}-\psi_{\xi}\frac{\partial}{\partial\eta})+\frac{(2\gamma-1)\psi\mu_{1}^{\prime}}{\gamma-1}\frac{\partial}{\partial\eta}.$

##  8 Isentropic case with $b_{01}^{2}+b_{02}^{2}=0$

First of all it should be notice that $\gamma=2$ is equivalent to the gas
dynamics equations. Hence, it is assumed that $\gamma\neq 2$. The defining
equations (6.1)-(6.4) reduce to the equations

$b_{03\eta}\xi^{\eta}+b_{03\xi}\xi^{\xi}=\frac{b_{03}}{\gamma-1}\left(k_{2}(\gamma-2)-\frac{\gamma-2}{2}k_{5}\right),$
(8.1) $\xi_{\eta}^{\eta}+\xi_{\xi}^{\xi}=\frac{1}{\gamma-1}(2\gamma
k_{2}+(\gamma-2)k_{5}).$ (8.2)

Assume that $b_{03}$ is not constant, for example, $b_{03\eta}\neq
0{\normalcolor{\normalcolor}}$. Finding $\xi^{\eta}$ from (8.1) and
substituting it into (8.2), one obtains a linear first-order partial
differential equation for the function $\xi^{\xi}$. Representing the general
solution of this equation in the form

$\xi^{\xi}=b_{03\eta}(\psi_{1}+\psi_{2}k_{2}+\psi_{3}k_{5}),$

one derives

$\psi_{1\xi}b_{03\eta}-\psi_{1\eta}b_{03\xi}=0,$
$\psi_{2\xi}b_{03\eta}-\psi_{2\eta}b_{03\xi}=\frac{(\gamma-2)b_{03}b_{03\eta\eta}}{(\gamma-1)b_{03\eta}^{2}}+\frac{(\gamma+2)}{(\gamma-1)},$
$\psi_{3\xi}b_{03\eta}-\psi_{3\eta}b_{03\xi}=\frac{(\gamma-2)}{2(\gamma-1)}\left(3-\frac{b_{03}b_{03\eta\eta}}{b_{03\eta}^{2}}\right).$

The extension of admitted Lie algebras occurs by the generators

$X_{9}^{(24)}=\psi_{1}\left(b_{03\eta}\frac{\partial}{\partial\xi}-b_{03\xi}\frac{\partial}{\partial\eta}\right),$
$X_{10}^{(24)}=\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+\psi_{2}\left(b_{03\eta}\frac{\partial}{\partial\xi}-b_{03\xi}\frac{\partial}{\partial\eta}\right)+\frac{(\gamma-2)b_{03}}{(\gamma-1)b_{03\eta}}\frac{\partial}{\partial\eta},$

$X_{11}^{(24)}=2\psi_{3}\left(b_{03\eta}\frac{\partial}{\partial\xi}-b_{03\xi}\frac{\partial}{\partial\eta}\right)-\frac{(\gamma-2)b_{03}}{(\gamma-1)b_{03\eta}}\frac{\partial}{\partial\eta}+\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+2t\frac{\partial}{\partial
t}.$

If $b_{03}$ is constant, then $k_{5}=2k_{2}$, $\xi^{\eta}=\psi_{1\xi}$,
$\xi^{\xi}=-\psi_{1\eta}+4k_{2}\xi$, where $\psi_{1}(\xi,\eta)$ is an
arbitrary function, and the extension of admitted Lie algebras occurs by the
generators

$X_{9}^{(25)}=\psi_{1\eta}\frac{\partial}{\partial\xi}-\psi_{1\xi}\frac{\partial}{\partial\eta},\,\,\,X_{10}^{(25)}=\varphi\frac{\partial}{\partial\varphi}+\zeta\frac{\partial}{\partial\zeta}+t\frac{\partial}{\partial
t}+2\xi\frac{\partial}{\partial\xi}.$

## Conclusions

The transition to Lagrangian coordinates allows integrating four equations of
magnetogasdynamics of an ideal perfect polytropic gas: the entropy
$S(\xi,\eta)$ and the functions associated with the magnetic field
$(b_{01}(\xi,\eta),b_{02}(\xi,\eta),b_{03}(\xi,\eta))$ are arbitrary functions
of the integration. This leads to complications in the study of group
classification: consideration of the many possibilities of these functions.
The analysis presented in this article gives a complete investigation of all
these possibilities. Figures 1-3 provide the trees of the study of
nonisentropic cases, where $(i,j)$ means the following: $i$ is the number of
the extension of the kernel of admitted Lie algebras (5.7), $j$ is the number
of the generators $X_{k+8}^{(i)},\ (k=1,2,...,j)$ in $i$th extension. Figure 4
presents the tree of the study for isentropic flows. The Lie algebras
corresponding to the extensions $i$ ($i=1,2,...,19$) are finite dimensional,
the Lie algebras corresponding to other extensions are infinite dimensional.

As mentioned above, finding an admitted Lie group is one of the first and
necessary steps in application of the group analysis method for constructing
invariant and partially invariant solutions. Because the equations (3.1) are
variational, the symmetries found can also be used to derive conservation laws
using Noether’s theorem. The wide variety of these symmetries allows us to
expect the derivation of new conservation laws. The search for invariant
solutions, as well as the derivation of conservation laws, are the subject of
further applications of the symmetries obtained in the present work.

## Acknowledgements

The research was supported by Russian Science Foundation Grant No. 18-11-00238
‘Hydrodynamics-type equations: symmetries, conservation laws, invariant
difference schemes’. E.I.K. acknowledges Suranaree University of Technology
(SUT) and Thailand Science Research and Innovation (TSRI) for Full-time
Doctoral Researcher Fellowship (Full-time61/15/2021).

$j_{1}\neq 0$$j_{2}=0$$N\neq 0$$\begin{array}[]{c}{P^{\prime\prime}\neq 0}\\\
\small{({10,1})}\end{array}$ $\begin{array}[]{c}{P^{\prime\prime}=0}\\\
\small{({11,2})}\end{array}$ $\begin{array}[]{c}{N=0}\\\
\small{({9,3})}\end{array}$ $j_{2}\neq 0$$j_{3}\neq
0$$\mu=0$$\nu=0$$\begin{array}[]{c}{M\neq 0}\\\ \small{({4,1})}\end{array}$
$\begin{array}[]{c}{M=0}\\\ \small{({3,2})}\end{array}$
$\begin{array}[]{c}{\nu\neq 0}\\\ \small{({2,1})}\end{array}$
$\begin{array}[]{c}{\mu\neq 0}\\\ \small{({1,1})}\end{array}$
$j_{3}=0$$\begin{array}[]{c}{j_{5}\neq 0}\\\ \small{(5,1),(6,2)}\end{array}$
$j_{5}=0$$\begin{array}[]{c}{\lambda=0}\\\ \small{({8,3})}\end{array}$
$\begin{array}[]{c}{\lambda\neq 0}\\\ \small{({7,2})}\end{array}$ Figure 1:
Tree of the study for $j_{1}\neq 0$, $b_{01}^{2}+b_{02}^{2}\neq 0$ and $S\neq
const$ $j_{1}=0$$g_{1}\neq 0$$\begin{array}[]{c}{g_{1}^{\prime}\neq 0}\\\
\small{({12,2})}\end{array}$ $\begin{array}[]{c}{g_{1}^{\prime}=0}\\\
\small{({13,3})}\end{array}$ $g_{1}=0$$\begin{array}[]{c}{g_{2\eta}\neq 0}\\\
\small{({14,2})}\end{array}$ $\begin{array}[]{c}{g_{2\eta}=0}\\\
\small{({16,3})}\end{array}$ Figure 2: Tree of the study for $j_{1}=0$,
$b_{01}^{2}+b_{02}^{2}\neq 0$ and $S\neq const$ $b_{01}=b_{02}=0$$f_{2}\neq
0$$\begin{array}[]{c}{f_{3}\neq const}\\\ \small{({17,1})}\end{array}$
$f_{3}=const$$\begin{array}[]{c}{f_{4}=const}\\\ \small{({18,2})}\end{array}$
$\begin{array}[]{c}{f_{4}\neq const}\\\ \small{({19,1})}\end{array}$
$f_{2}=0$$\begin{array}[]{c}{f_{5}=const}\\\ \small{({21,3})}\end{array}$
$\begin{array}[]{c}{f_{5}\neq const}\\\ \small{({20,2})}\end{array}$ Figure 3:
Tree of the study for $b_{01}=0$, $b_{02}=0$ and $S\neq const$
$S=const$$b_{01}^{2}+b_{02}^{2}\neq 0$$\begin{array}[]{c}{\psi_{\eta\eta}\neq
0}\\\ \small{({22,4})}\end{array}$ $\begin{array}[]{c}{\psi_{\eta\eta}=0}\\\
\small{({23,3})}\end{array}$
$b_{01}^{2}+b_{02}^{2}=0$$\begin{array}[]{c}{b_{03\eta}\neq 0}\\\
\small{({24,3})}\end{array}$ $\begin{array}[]{c}{b_{03}=const}\\\
\small{({25,2})}\end{array}$ Figure 4: Tree of the study for isentropic flows
$S=const$.

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and $B(r,\varepsilon^{\prime})=E(r,\varepsilon^{\prime})\cap C.$ We want to
show that there exists $\varepsilon^{\prime}>0$ such that if
$\varepsilon^{\prime}<1$ then
$\mathbb{P}[E(1,\varepsilon^{\prime})]\leq\delta.$

Note that if $B(1,\varepsilon^{\prime})$ holds, then for all $A\in\mathcal{A}$

$\displaystyle\left|\frac{\sum_{i\in
I_{\operatorname{cal},r}(A)}\mathbbm{1}\left\\{\widehat{s}_{\operatorname{train}}(X_{i},Y_{i})\leq
q_{\operatorname{train}}\right\\}}{\\#I_{\operatorname{cal},r}(A)}-P_{q,\operatorname{train}}(A)\right|>\varepsilon^{\prime}-\frac{2(r-1)}{\gamma
n_{\operatorname{cal}}}.$

That is, $B(1,\varepsilon^{\prime})\subset
B\left(r,\varepsilon^{\prime}-\frac{2(r-1)}{\gamma
n_{\operatorname{cal}}}\right)$. Now, define

$\mathbb{P}_{*}=\mathbb{P}_{1}^{n_{\operatorname{train}}}\otimes\mathbb{P}_{n_{\operatorname{train}}+r}^{n_{\operatorname{train}}+n_{\operatorname{cal}}},$

so under $\mathbb{P}_{*}$ we have that $(X_{i},Y_{i})_{i\in
I_{\operatorname{train}}}$ and $(X_{i},Y_{i})_{i\in I_{\operatorname{cal}},r}$
are independent. By Proposition B.1 we have

$\displaystyle\mathbb{P}[B(1,\varepsilon^{\prime})]$
$\displaystyle\leq\mathbb{P}\left[B\left(r,\varepsilon^{\prime}-\frac{2(r-1)}{\gamma
n_{\operatorname{cal}}}\right)\right]$
$\displaystyle\leq\mathbb{P}_{*}\left[B\left(r,\varepsilon^{\prime}-\frac{2(r-1)}{\gamma
n_{\operatorname{cal}}}\right)\right]+\beta(r).$

But this implies that

$\mathbb{P}[E(1,\varepsilon^{\prime})]\leq\mathbb{P}_{*}\left[B\left(r,\varepsilon^{\prime}-\frac{2(r-1)}{\gamma
n_{\operatorname{cal}}}\right)\right]+\beta(r)+1-\mathbb{P}[C].$

For any $m,a\in\mathbb{N}_{+}$ with $n_{\operatorname{cal}}-r+1=2ma$ and
$\delta_{\operatorname{cal}}>8(m-1)\beta(a)$, if we take

$\displaystyle\varepsilon^{\prime}=\frac{1}{\gamma}\left(4\sqrt{\frac{\log(2(m+1)^{d})}{m}}+2\sqrt{\frac{1}{2m}\log\left(\frac{8}{\delta-8(m-1)\beta(a)}\right)}+\frac{2(r-1)}{n_{\operatorname{cal}}}\right),$

and assume that $\varepsilon^{\prime}<1$ then

$\frac{1}{\gamma}\left(4\sqrt{\frac{\log(2(m+1)^{d})}{m}}+2\sqrt{\frac{1}{2m}\log\left(\frac{8}{\delta-8(m-1)\beta(a)}\right)}\right)<1$

so Lemma C.7 tells us that

$\mathbb{E}_{*}\left[\mathbb{P}_{*}\left[B\left(r,\varepsilon^{\prime}-\frac{2(r-1)}{\gamma
n_{\operatorname{cal}}}\right)\mid(X_{i},Y_{i})_{i\in
I_{\operatorname{train}}}\right]\right]\leq\delta$

and Lemma C.6 tells us $1-\mathbb{P}[C]\leq\delta$. That is,

$\mathbb{P}[E(1,\varepsilon^{\prime})]\leq 2\delta+\beta(r),$

which is equivalent to say that

$\mathbb{P}[E(1,\varepsilon)]\leq\delta,$

if $\varepsilon$ is as in the proposition statement. ∎

######  Proposition C.9.

Assumption 7 holds with

$\varepsilon_{\operatorname{train}}=\beta(k-n_{\operatorname{train}}).$

###### Proof.

Given $k\in I_{\operatorname{test}}$, note that we can decompose

$\mathbb{P}_{*}=\mathbb{P}_{1}^{n_{\operatorname{train}}}\otimes\mathbb{P}_{k}^{k},$

so under $\mathbb{P}_{*}$ we have that $(X_{k},Y_{K})$ is independent of
$(X_{i},Y_{i})_{i\in I_{\operatorname{train}}}.$ Then, defining
$\beta_{k}=\beta(k-n_{\operatorname{train}})$ we have for all
$A\in\mathcal{A},$

$\displaystyle\beta_{k}$
$\displaystyle\geq\left|\mathbb{P}[\widehat{s}_{\operatorname{train}}(X_{k},Y_{k})\leq{q}_{\operatorname{train}}(A),X_{k}\in
A]-\mathbb{P}_{*}[\widehat{s}_{\operatorname{train}}(X_{k},Y_{k})\leq{q}_{\operatorname{train}}(A),X_{k}\in
A]\right|$

where the $\beta_{k}$ penalty follows from Proposition B.1. But then, by a
conditioning argument,

$\displaystyle\beta_{k}$
$\displaystyle\geq\left|\mathbb{P}[\widehat{s}_{\operatorname{train}}(X_{k},Y_{k})\leq{q}_{\operatorname{train}}(A),X_{k}\in
A]-\mathbb{P}[\widehat{s}_{\operatorname{train}}(X_{*},Y_{*})\leq{q}_{\operatorname{train}}(A),X_{*}\in
A]\right|.$

Finally, dividing by $\mathbb{P}[X_{k}\in A]=\mathbb{P}[X_{*}\in A]$ and using
the fact that

$\frac{\beta_{k}}{\mathbb{P}[X_{k}\in A]}\geq\beta_{k}$

yields the result. ∎

######  Proposition C.10.

Define

$\displaystyle\varepsilon=\inf_{(a,m,s)\in
G_{\operatorname{test}}}\left\\{\frac{2}{\gamma}\left(\varepsilon_{0}\left(a,m,\frac{\delta-\beta(n_{\operatorname{cal}})}{2}\right)\right)+\frac{s}{n_{\operatorname{test}}}\right\\}$

where

$G_{\operatorname{test}}=\\{(a,m)\in\mathbb{N}_{>0}^{2}\ :\
s+2ma=n_{\operatorname{test}},\delta>8(m-1)\beta(a)+\beta(n_{\operatorname{cal}})\\}.$

If $\varepsilon<1$, then Assumption 8 holds with
$\varepsilon_{test}=\varepsilon.$

###### Proof.

The proof is similar to the proof of Proposition C.8. Let the event
$E(\varepsilon)$ be

$E(\varepsilon)=\left\\{\inf_{A\in\mathcal{A}}\left|P_{q,\operatorname{train}}(A)-\frac{\sum_{i\in
I_{\operatorname{test}}(A)}\mathbbm{1}\left\\{\widehat{s}_{\operatorname{train}}(X_{i},Y_{i})\leq
q_{\operatorname{train}}]\right\\}}{n_{\operatorname{test}}(A)}\right|>\varepsilon\right\\},$

Define,

$\mathbb{P}_{*}=\mathbb{P}_{1}^{n_{\operatorname{train}}}\otimes\mathbb{P}_{n_{\operatorname{train}}+n_{\operatorname{cal}}}^{n_{\operatorname{train}}+n_{\operatorname{cal}}+n_{\operatorname{test}}},$

so under $\mathbb{P}_{*}$ we have that $(X_{i},Y_{i})_{i\in
I_{\operatorname{train}}}$ and $(X_{i},Y_{i})_{i\in I_{\operatorname{test}}}$
are independent. By Proposition B.1 we have

$\displaystyle\mathbb{P}[E(\varepsilon)]$
$\displaystyle\leq\mathbb{P}_{*}[E(\varepsilon)]+\beta(n_{\operatorname{cal}})$
$\displaystyle=\mathbb{E}_{*}[\mathbb{P}_{*}[E(\varepsilon)\mid(X_{i},Y_{i})_{i\in
I_{\operatorname{train}}}]]+\beta(n_{\operatorname{cal}}).$

Now we can apply Lemma C.7 and conclude that if $\varepsilon<1$, for any
$m,a\in\mathbb{N}_{+}$ with $n_{\operatorname{test}}=2ma$ and
$\delta_{\operatorname{test}}>8(m-1)\beta(a))+\beta(n_{\operatorname{cal}})$,
it is true that $\mathbb{P}[E]\leq\delta$, where

$\varepsilon=\frac{1}{\gamma}\left(4\sqrt{\frac{\log(2(m+1)^{d})}{m}}+2\sqrt{\frac{1}{2m}\log\left(\frac{8}{\delta-8(m-1)\beta(a)-\beta(n_{\operatorname{cal}})}\right)}+\frac{s}{n_{\operatorname{test}}}\right).$

Finally, since this is true for any choice of $a,m,s\in\mathbb{N}_{>0}$ with
$s+2ma=n_{\operatorname{test}}$ and
$\delta_{\operatorname{test}}>8(m-1)\beta(a)+\beta(n_{\operatorname{cal}})$,
we can choose $a,m,s$ optimally such that the value of $\varepsilon$ is
minimized. ∎

##  Appendix D Proofs of Section 4

###### Proof of Theorem 4.2.

This proof is similar to the proof of Theorem 2.3. Indeed, if we consider
$I_{\operatorname{cal}}\coloneqq I_{\operatorname{test}}$ in the proof of
Theorem 2.3, the event

$F=\\{q_{1-\alpha-\varepsilon_{\operatorname{cal}},\operatorname{train}}\leq\widehat{q}_{1-\alpha,\operatorname{cal}}\\},$

satisfies $\mathbb{P}[F]\geq 1-\delta_{\operatorname{cal}}.$ But since,

$\widehat{q}^{(i)}_{1-\alpha,\operatorname{cal}}\geq\widehat{q}_{1-\alpha-1/n_{\operatorname{test}},\operatorname{cal}},$

the following event

$F^{\prime}=\\{q_{1-\alpha-\varepsilon_{\operatorname{cal}}-1/n_{\operatorname{test}},\operatorname{train}}\leq\widehat{q}^{(i)}_{1-\alpha,\operatorname{cal}}\\},$

also satisfies $\mathbb{P}[F^{\prime}]\geq 1-\delta_{\operatorname{cal}}$. The
rest of the proof follows the same strategy as in Theorem 2.3 using
$\widehat{q}^{(i)}_{1-\alpha,\operatorname{cal}}$ instead of
$\widehat{q}_{1-\alpha,\operatorname{cal}}$. ∎