Integrability analysis of a simple model for describing convection of a rotating fluid
Jia Jiao
College of Science
Dalian Minzu University
Dalian 116600
P R China
E-mail: jiaojia@dlnu.edu.cn
Qingjian Zhou
College of Science
Dalian Minzu University
Dalian 116600
P R China
E-mail: zqj68@dlnu.edu.cn
Shuangling Yang
School of Mathematics
Jilin University
Changchun 130012
P R China
E-mail: yangsl19@mails.jlu.edu.cn
We study the Darboux integrability of a simple system of three ordinary differential equations called the Glukhovsky-Dolzhansky system, which describes a three-mode model of rotating fluid convection inside the ellipsoid. (1) Our results show that it has no polynomial, rational, or Darboux first integrals for any value of parameters in the physical sense, that is, positive parameters. (2) We also provide some integrable cases of this model when parameters are allowed to be non-positive. (3) We finally give some links between the Glukhovsky-Dolzhansky system and other similar systems in $\mathbb{R^3$, which admits rotational symmetry and has three nonlinear cross terms.
ss1.tex
%                          EXAMPLE  (LaTeX2e)
\documentclass{article}
\usepackage{romp}
\usepackage{graphicx}
%% The amssymb package provides various useful mathematical symbols
\usepackage{amssymb}
%% The amsthm package provides extended theorem environments
%% \usepackage{amsthm}

%% The lineno packages adds line numbers. Start line numbering with
%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
%% for the whole article with \linenumbers after \end{frontmatter}.
\usepackage{lineno}
\usepackage{mathrsfs}
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{indentfirst}
\usepackage{color}
\usepackage{changepage}
\usepackage{epsfig,psfrag}


\usepackage{graphicx}
\usepackage{epstopdf}

\usepackage{subfigure}

\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{float}
\usepackage{subfigure}





\title{Integrability analysis of a simple model for describing  convection of a rotating fluid }
\author{ Jia Jiao
\\ College of Science, Dalian Minzu University, Dalian 116600, P R China
\\ E-mail: jiaojia@dlnu.edu.cn
\\ Qingjian Zhou
\\ College of Science, Dalian Minzu University, Dalian 116600, P R China
\\ E-mail: zqj68@dlnu.edu.cn
\\ Shuangling Yang 
\\ School of Mathematics, Jilin University,  Changchun 130012, P R China
\\ E-mail: yangsl19@mails.jlu.edu.cn}

\begin{document}

\maketitle
\begin{abstract}
 We study the Darboux integrability of a simple system of three ordinary differential equations
called the Glukhovsky-Dolzhansky system, which describes a three-mode model of rotating fluid convection inside the ellipsoid.
(1) Our results show that  it has no  polynomial, rational, or Darboux first integrals for any value of parameters in the physical sense, that is, positive parameters. (2) We also provide some integrable cases of this model when parameters are allowed to be non-positive. (3)
We finally give some links between the Glukhovsky-Dolzhansky system and other similar systems in $\mathbb{R}^3$, which admits rotational symmetry and has three nonlinear cross terms.
\end{abstract}

\noindent
{\bf Keywords:} Glukhovsky-Dolzhansky system; Darboux integrability;  Darboux polynomials; Exponential factor
\footnote{S. Yang, corresponding author}
\section{Introduction}

Consider the convection of viscous incompressible fluid motion inside the ellipsoid
\begin{equation*}\label{I0}
(\frac{x_1}{a_1})^{2}+(\frac{x_2}{a_2})^{2}+(\frac{x_3}{a_3})^{2}=1,~~~~a_1>a_2>a_3>0,
\end{equation*}
under the condition of stationary inhomogeneous external heating.
Glukhovsky and Dolzhansky \cite{model} assumed that the ellipsoid rotates with the constant velocity $\Omega_0$ along the axis $a_3$,
and the axis $a_3$ has a constant angle $\alpha$ with the gravity vector $g$.
They also assumed that the temperature difference is generated along the axis $a_1$ and its gradient has a constant value $q_0$.
Denote by $\lambda,\mu,\beta$ the coefficients of viscosity,
heat conduction,
and volume expansion,
respectively.
Then they proposed the following system of ordinary differential equations
\begin{equation}\label{I1}
\begin{cases}
\dot{x}=Ayz+Cz-\sigma x:=P(x,y,z),\\
\dot{y}=-xz+Ra-y:=Q(x,y,z),\\
\dot{z}=-z+xy:=R(x,y,z),
\end{cases}
\end{equation}
which can be
interpreted as one of the models of ocean flows.
Here
\begin{align*}
&\sigma=\frac{\lambda}{\mu},~~~T_{\alpha}=\frac{\Omega_{0}^{2}}{\lambda^{2}},
~~~R_{a}=\frac{g\beta a_{3}q_{0}}{2a_{1}a_{2}\lambda\mu},\\
&A=\frac{a_{1}^{2}-a_{2}^{2}}{a_{1}^{2}+a_{2}^{2}}\cos^{2}\alpha T_{\alpha}^{-1},~~
C=\frac{a_{1}^{2}-a_{2}^{2}}{a_{1}^{2}+a_{2}^{2}}\cos^{2}\alpha T_{\alpha}^{-1},\\
&x(t)=\mu^{-1}\omega_{3}(t),~~y(t)=\frac{g\beta a_{3}}{2a_{1}a_{2}\lambda \mu}q_{1}(t),~~z(t)=\frac{g\beta a_{3}}{2a_{1}a_{2}\lambda \mu}q_{2}(t),
\end{align*}
$\omega_3(t)$ is the projection of the vector of angular velocity  on the axis $a_3$,   $q_{1}(t)$ and $q_{2}(t)$ are the projections of temperature gradients on the axes $a_1$ and $a_2$.
The parameters $\sigma$, $T_{\alpha}$ and $R_{a}$ are the Prandtl,
Taylor and Rayleigh numbers, respectively.
Clearly,
the parameters $A,C,\sigma$ and $Ra$ of system (\ref{I1}) are positive.








In spite of its simple form, the Glukhovsky-Dolzhansky system (\ref{I1}) admits rich dynamics and has been intensively investigated,
see \cite{gb0,gb1,gb2,gb3} and
the references therein.
However,
this system has never been studied from the integrability point of view.
The main purpose of this work is to cover this gap and to investigate the integrability of system (\ref{I1}).
The investigation of integrability for systems of differential equations has been one of significant topics not
only in mathematics but also in physics.
Generally, a system of differential equations is integrable if it possesses a sufficient number of first integrals (and/or other tensor invariants) such that we can solve this system explicitly.
Hence we could obtain its global information and understand its topological structure \cite{dd2,dd3}.
Furthermore,
non-integrability of the system also seems necessary for better understanding of the complex phenomenon \cite{in1,in3}.
However, to study the integrability of
a given system is not an easy task,
since there is no any effective approach to determine the existence or nonexistence of first integrals.





\begin{figure}[htpp]
\centering
\subfigure[]{
\label{Fig.sub.1}
\includegraphics[width=0.50\textwidth]{xyz.eps}}
\subfigure[]{
\label{Fig.sub.2}
\includegraphics[width=0.45\textwidth]{xy.eps}}
\caption{(a) chaotic attractor for the system (\ref{I1}) with parameter values $(A,C,\sigma,Ra)=(0.04,1,4,250)$; (b) its projections onto the planes $(x, y)$.}
\label{Fig.lable}
\end{figure}

\begin{figure}[htpp]
\centering
\subfigure[]{
\label{Fig.sub.1}
\includegraphics[width=0.45\textwidth]{xz.eps}}
\subfigure[]{
\label{Fig.sub.2}
\includegraphics[width=0.45\textwidth]{yz.eps}}
\caption{Projections of the chaotic attractor into the plane $(x ,z)$ (a) and into the plane $(y, z)$ (b).}
\label{Fig.lable}
\end{figure}

In this paper,
we aim to study the integrability of system (\ref{I1}) in the framework of Darboux integrability theory.
Darboux integrability theory plays an important role in the integrability of
the polynomial differential systems \cite{dar1,dar2,dar3,dar4,dar5},
which helps us find first integrals by knowing a sufficient number of algebraic
invariant surfaces (the Darboux polynomials) and of the exponential factors,
see \cite{d1,d2,d3,d4,d5} for instance.
Moreover,
it can also help us make a more precise analysis of the global dynamics of the considered system topologically \cite{dd1,dd2}.
In addition, let us mention that the Darboux integrability of  the Rabinovich system, 3D forced-damped system
and \emph{D2} vector field,
which are similar but different from  the Glukhovsky-Dolzhansky system (\ref{I1}),  have been studied in \cite{si1,si2,si3,si4}.
These systems  admit a common structure: the original is
the equilibrium point, symmetry with respect to certain coordinate axis, only three cross nonlinear terms $\{yz,xz,xy\}$.
In the appendix, we provide some linear rescaling of time and coordinate to show the Glukhovsky-Dolzhansky system (\ref{I1}) can be transformed into the
Rabinovich system or 3D forced-damped system or \emph{D2} vector field  only when the parameters satisfies some conditions: $C=-2AR_a$ or $\sigma=1$ or $Ra=C=0$ respectively.



The main result of this paper is as following.
\begin{theorem}{Theorem}\label{th}
The following statements hold for the Glukhovsky-Dolzhansky system (\ref{I1}).\\
(a.) It has no polynomial first integrals.\\
(b.) It has no Darboux polynomials with non-zero cofactors.\\
(c.) It has no exponential factors.\\
(d.) It admits no Darboux first integrals.
\end{theorem}

\begin{remark}{Remark}
The Glukhovsky-Dolzhansky system admits complex dynamics for a larger range of its parameters. For example, it has a
strange attractor with  $(A,C,\sigma,Ra)=(0.04,1,4,250)$, see Fig 1 and 2. Our non-integrability results coincide with this fact.
As mentioned above, from a physical point of view, the parameters in the Glukhovsky-Dolzhansky system (\ref{I1}) is positive
due to their  physical meaning.  Whereas, from a mathematical point of view, one may consider non-positive parameters and then
some integrable cases of system (\ref{I1}) are founded:
\begin{itemize}
\item $Ra=0$ and $\sigma=1$, system (\ref{I1}) has a Darboux polynomial $f=y^2+z^2$ with a constant cofactor $k=-2$.
\item $C=0$ and $\sigma=1$, system (\ref{I1}) has a Darboux polynomial $f=x^2-Az^2$ with a constant cofactor $k=-2$.
\item $Ra=C=0$ and $\sigma=1$, system (\ref{I1}) has a rational first integral $\Phi=(y^2+z^2)/(x^2-Az^2)$.
\end{itemize}
But we do not have a clear physical understanding of the above integrable results for the Glukhovsky-Dolzhansky system (\ref{I1}).
\end{remark}
%However, if we allow that the value of  parameters doesn't make physical sense. For example, supposing $(A,C,\sigma,Ra)=(1,0,)$, we see that system (\ref{I1})
\begin{remark}{Remark}
Another tool to study the non-integrability of non-Hamiltonian systems is  the differential Galois theory \cite{ga1,ga2,ga3}.
Observing system (\ref{I1}) has a straight line solution $(x(t),y(t),z(t))=(0,Ra-e^{-t},0)$,
we can analyze the differential
Galois group of the normal variational equations along this solution,
and show that system (\ref{I1}) is not rationally integrable in Bogoyavlenskij sense for almost all parameter values, see \cite{gaa1,gaa2,gaa3,newgaa3}
for more details.
\end{remark}


The paper is organized as follows.
In Section 2 we introduce some preliminary notions and results related to Darboux
theory of integrability that we will use later.
The proof of Theorem \ref{th} is given in Section 3.



\section{Preliminary results}

Let $\mathbb{R}[x,y,z]$ be the ring of the real polynomials in the variables $x$,
$y$ and $z$. We say that $f(x,y,z)\in \mathbb{R}[x,y,z]$ is a \emph{Darboux polynomial} of system (\ref{I1}) if it satisfies
\begin{equation}\label{P1}
\frac{\partial f}{\partial x}P+\frac{\partial f}{\partial y}Q+\frac{\partial f}{\partial z}R=Kf,
\end{equation}
for some polynomial $K$, called the cofactor of $f(x,y,z)$.
If $f(x,y,z)$ is a Darboux polynomial,
then the surface $f(x,y,z)=0$ is an invariant manifold of system (\ref{I1}).
Particularly, if $K=0$, $f(x,y,z)$ satisfies the following equation
\begin{equation}\label{P2}
\frac{\partial f}{\partial x}P+\frac{\partial f}{\partial y}Q+\frac{\partial f}{\partial z}R=0,
\end{equation}
then polynomial $f(x,y,z)$ is called a \emph{polynomial first integral} of system (\ref{I1}) .

Let $g,h\in \mathbb{R}[x,y,z]$ be coprime. We say that a nonconstant function $E=\exp(g/h)$ is an \emph{exponential factor} of system (\ref{I1})
if $E$ satisfies
\begin{equation*}\label{P3}
\frac{\partial E}{\partial x}P+\frac{\partial E}{\partial y}Q+\frac{\partial E}{\partial z}R=LE,
\end{equation*}
for some polynomial $L\in\mathbb{R}[x,y,z]$ with the degree at most one, called the cofactor of $E$.

A first integral $G$ of system (\ref{I1}) is called  \emph{Darboux type} if it is a first integral of the form
\begin{equation*}\label{P4}
G=f_{1}^{\lambda_{1}}\cdots f_{p}^{\lambda_{p}}E_{1}^{\mu_{1}}\cdots E_{q}^{\mu_{q}},
\end{equation*}
where $f_{1},\cdots,f_{p}$ are Darboux polynomials,
$E_{1},\cdots,E_{q}$ are exponential factors and $\lambda_{i},\mu_{j}\in R$,
for $i=1,\cdots,p$ and $j=1,\cdots,q$.


To prove Theorem \ref{th}, we need the following results.
\begin{proposition}{Proposition}\label{PP1}
Suppose that a polynomial vector field $X$ defined in $\mathbb{R}^n$ of degree
$m$ admits $p$ Darboux polynomials $f_{i}$ with cofactor $K_{i}$ for $i=1,\cdots,p,$
and $q$ exponential factors $E_{j}=\exp(g_{j}/h_{j})$ with cofactors $L_{j}$ for
$j=1,\cdots,q$. If there exist $\lambda_{i}$, $\mu_{j}\in\mathbb{R}$ not all zero
such that
\begin{equation*}\label{P5}
\sum_{i=1}^{p}\lambda_{i}K_{i}+\sum_{j=1}^{q}\mu_{j}L_{j}=0,
\end{equation*}
then the following real (multivalued) function of Darboux type
\begin{equation*}\label{P6}
f_{1}^{\lambda_{1}}\cdots f_{p}^{\lambda_{p}}E_{1}^{\mu_{1}}\cdots E_{p}^{\mu_{q}},
\end{equation*}
substituting $f_{i}^{\lambda_{i}}$ by $|f_{i}|^{\lambda_{i}}$ if $\lambda_{i}\in\mathbb{R}$,
is a first integral of the vector field $X$.
\end{proposition}

The proof can be seen in \cite{dar1}.

\begin{proposition}{Proposition}\label{PP2}
The following statements hold.\\
(a) If $e^{g/h}$ is an exponential factor for the polynomial differential system (\ref{I1})
and $h$ is not a constant polynomial, then $h$ is a Darboux polynomial\\
(b) Eventually $e^{g}$ can be an exponential factor, coming from the multiplicity of the
infinite invariant plane.
\end{proposition}

For a proof of this result see \cite{dar2}.

\begin{proposition}{Proposition}\label{PP3}
Let $f$ be a polynomial and $f=\prod_{j=1}^{s}f_{j}^{\alpha_{j}}$ be its decomposition into irreducible
factors in $\mathbb{R}[x,y,z]$. Then $f$ is a Darboux polynomial of system (\ref{I1}) if and only if all
the $f_{j}$ are Darboux polynomials of system (\ref{I1}). Moreover, if $K$ and $K_{j}$ are the cofactors of
$f$ and $f_{j}$, then $K=\sum_{j=1}^{s}\alpha_{j}K_{j}$.
\end{proposition}

The proof of Proposition \ref{PP3} can be found in \cite{dar3}.

\section{Proof of Theorem \ref{th}}

We separate the proof of Theorem \ref{th} into the following different propositions.
\begin{proposition}{Propositon}\label{T1}
System (\ref{I1})  has no polynomial first integrals.
\end{proposition}
{\bf Proof:}  Suppose
\begin{align}\label{poly}
f(x,y,z)=\sum_{i=0}^{n}f_{i}(x,y,z)
\end{align}
is a polynomial first integral of system (\ref{I1}), where $f_i$ are the homogeneous polynomials of degree $i$ and $f_{n}\neq0$.
Firstly, substituting $(\ref{poly})$ into (\ref{P2}) and identifying the homogeneous components of degree $n+1$, we get
\begin{equation}\label{po}
Ayz\frac{\partial f_{n}}{\partial x}-xz\frac{\partial f_n}{\partial y}+xy\frac{\partial f_n}{\partial z}=0.
\end{equation}
The characteristic equations associated with (\ref{po}) are
\begin{equation*}\label{D5}
\frac{dx}{dy}=\frac{Ayz}{-xz},~~\frac{dz}{dy}=\frac{xy}{-xz}.
\end{equation*}
Its general solution are
\begin{equation*}\label{D6}
x^{2}+Ay^{2}=c_{1},~~y^{2}+z^{2}=c_{2},
\end{equation*}
where $c_{1}$ and $c_{2}$ are arbitrary constants.
We make the change of variables
\begin{equation*}\label{D7}
u=x^{2}+Ay^{2},~~w=y,~~v=y^{2}+z^{2}.
\end{equation*}
Correspondingly, the inverse transformation is
\begin{equation}\label{D8}
x=\pm\sqrt{u-Aw^{2}},~~y=w,~~z=\pm\sqrt{v-w^{2}}.
\end{equation}
Without loss of generality, we set
\begin{equation}\label{D9}
x=\sqrt{u-Aw^{2}},~~y=w,~~z=-\sqrt{v-w^{2}},
\end{equation}
and transform (\ref{po}) into
\begin{align}\label{D12}
\frac{d\bar{f}_n}{d w}=0,
\end{align}
where $\bar{f}_n(u,v,w)=f_n(x,y,z)$. In the following, unless otherwise specified,
we always denote the function $R(x,y,z)$ by $\bar{R}(u,v,w)$.
Hence we obtain
\begin{equation*}\label{D14}
f_n(x,y,z)=f_{2m}(x,y,z)=\sum_{i=0}^{m}a_i^m(x^{2}+Ay^{2})^{m-i}(y^{2}+z^{2})^{i},~~a_i^m\in\mathbb{R},
\end{equation*}
where $n=2m$ must be an even number.
Secondly, substituting $(\ref{poly})$ into (\ref{P2}) and identifying the homogeneous components of degree $n$ yields
\begin{align}
Ayz\frac{\partial f_{n-1}}{\partial x}&-xz\frac{\partial f_{n-1}}{\partial y}+xy\frac{\partial f_{n-1}}{\partial z}
=(\sigma x-Cz)\frac{\partial f_{n}}{\partial x}+y\frac{\partial f_{n}}{\partial y}+z\frac{\partial f_{n}}{\partial z}\label{assd}\\
=&\sum_{i=0}^{m}a_{i}^{m}2\sigma(m-i)(x^2+Ay^2)^{m-i-1}(y^2+z^2)^ix^2\notag\\
&-\sum_{i=0}^{m}a_{i}^{m}2C(m-i)(x^2+Ay^2)^{m-i-1}(y^2+z^2)^ixz\notag\\
&+\sum_{i=0}^{m}a_{i}^{m}2A(m-i)(x^2+Ay^2)^{m-i-1}(y^2+z^2)^{i}y^{2}\notag\\
&+\sum_{i=0}^{m}a_{i}^{m}2i(x^2+Ay^2)^{m-i}(y^2+z^2)^{i-1}y^{2}\notag\\
&+\sum_{i=0}^{m}a_{i}^{m}2i(x^2+Ay^2)^{m-i}(y^2+z^2)^{i-1}z^{2}.\notag
\end{align}
Using the transformations (\ref{D9}) again, the above equation becomes
\begin{align*}
\sqrt{u-Aw^2}\sqrt{v-w^2}\frac{d\bar{f}_{2m-1}}{dw}=&\sum_{i=0}^{m}a_{i}^{m}2\sigma(m-i)u^{m-i-1}v^i(u-Aw^2)\\
&+\sum_{i=0}^{m}a_{i}^{m}2C(m-i)u^{m-i-1}v^i\sqrt{u-Aw^2}\sqrt{v-w^2}\\
&+\sum_{i=0}^{m}a_{i}^{m}2A(m-i)u^{m-i-1}v^{i}w^{2}+\sum_{i=0}^{m}a_{i}^{m}2iu^{m-i}v^{i-1}w^{2}\\
&+\sum_{i=0}^{m}a_{i}^{m}2iu^{m-i}v^{i-1}(v-w^2).
\end{align*}
We obtain
\begin{align*}
\bar{f}_{2m-1}
=&\sum_{i=0}^{m}a_{i}^{m}2\sigma(m-i)u^{m-i-1}v^i\int\frac{\sqrt{u-Aw^{2}}}{\sqrt{v-w^2}}dw\\
&+\sum_{i=0}^{m-1}2[a_{i}^{m}A(m-i)+a_{i+1}^{m}(i+1)]u^{m-i-1}v^{i}\int\frac{w^{2}dw}{\sqrt{u-Aw^{2}}\sqrt{v-w^2}}\\
&+\sum_{i=0}^{m}a_{i}^{m}2iv^{i-1}\int\frac{\sqrt{v-w^{2}}}{\sqrt{u-Aw^2}}dw\\
&+\sum_{i=0}^{m}a_{i}^{m}2C(m-i)u^{m-i-1}v^iw+A_{n-1}(u,v),
\end{align*}
where $A_{n-1}(u,v)$  is an arbitrary smooth function in $u$ and $v$.
It is easy to check
\begin{eqnarray}\label{D15}
\int\frac{w^2dw}{\sqrt{u-Aw^2}\sqrt{v-w^2}}
=-\int\frac{\sqrt{u-Aw^2}}{\sqrt{v-w^2}}d w
+u\int\frac{d w}{\sqrt{u-Aw^2}\sqrt{v-w^2}}.
\end{eqnarray}
%where
%\begin{eqnarray*}
%\int\frac{w d w}{\sqrt{u-Aw^2}\sqrt{v-w^2}}
%=\frac{1}{2\sqrt{A}}\ln\frac{2Aw^{2}-Av+2\sqrt{(u-Aw^{2})(v-w^{2})}\sqrt{A}-u}{2\sqrt{A}},
%\end{eqnarray*}
Since
\begin{align*}
\int\frac{\sqrt{u-Aw^2}}{\sqrt{v-w^2}}dw,~~~~~
\int\frac{ d w}{\sqrt{u-Aw^2}\sqrt{v-w^2}}d w
\end{align*}
are elliptic integrals of the second and first kinds, respectively,
in order that $f_{2m-1}(x,y,z) =\bar f_{2m-1}(u,v,w)$ is a homogeneous polynomial of degree $2m-1$, we must have
\begin{equation}\label{D16}
\begin{cases}
a_{i}^{m}2\sigma(m-i)=0,~~~i=0,1,\cdots,m\\
a_{i}^{m}2C(m-i)=0, ~~~i=0,1,\cdots,m\\
a_{i}^{m}2i=0, ~~~i=0,1,\cdots,m\\
a_{i}^{m}A(m-i)+a_{i+1}^{m}(i+1)=0, ~~~i=0,1,\cdots,m-1.
\end{cases}
\end{equation}
By (\ref{D16}) and $A,C,\sigma$ being positive,
we obtain $a_{i}^{m}=0, i=0,\cdots,m$.
This leads to a contradiction.
So system (\ref{I1})  has no polynomial first integrals.



\begin{proposition}{Proposition}\label{T2}
System (\ref{I1}) has no Darboux polynomial with  nonzero cofactors.
\end{proposition}
{\bf Proof:}
Suppose
\begin{align}\label{aa1}
f(x,y,z)=\sum_{i=0}^{n}f_{i}(x,y,z)
\end{align}
is a Darboux polynomial of the system (\ref{I1}) with a non-cofactor $K(x,y,z)$, where $f_{i}$ is a homogeneous polynomial of degree $i$ for $i=0,1,\cdots,n$. Comparing the degree on both sides of (\ref{P1}) yields $\deg K\leq 1$.
Without loss of generality,
we can assume that the cofactor is of the form
\begin{equation}\label{DD2}
K(x,y,z)=k_1x+k_2y+k_3z+k_0,~~k_i\in\mathbb{R},~~i=0,1,2,3.
\end{equation}
Substituting (\ref{aa1}) and $(\ref{DD2})$ into  (\ref{P1}) and identifying the terms of the same degree,
we obtain
\begin{align}\label{DD3}
Ayz\frac{\partial f_{n}}{\partial x}-xz\frac{\partial f_n}{\partial y}+xy\frac{\partial f_n}{\partial z}&=(k_1x+k_2y+k_3z)f_n,\\\nonumber
Ayz\frac{\partial f_{n-1}}{\partial x}-xz\frac{\partial f_{n-1}}{\partial y}+xy\frac{\partial f_{n-1}}{\partial z}=&(k_1x+k_2y+k_3z)f_{n-1}+(\sigma x-Cz)\frac{\partial f_{n}}{\partial x}\\\nonumber
&+y\frac{\partial f_{n}}{\partial y}
+z\frac{\partial f_{n}}{\partial z}+k_0f_{n},\\\label{DD4}
Ayz\frac{\partial f_{i}}{\partial x}-xz\frac{\partial f_i}{\partial y}+xy\frac{\partial f_i}{\partial z}=&(k_1x+k_2y+k_3z)f_i+(\sigma x-Cz)\frac{\partial f_{i+1}}{\partial x}\\\nonumber
&+y\frac{\partial f_{i+1}}{\partial y}
+z\frac{\partial f_{i+1}}{\partial z}-Ra\frac{\partial f_{i+2}}{\partial y}+k_0f_{i+1}, \\
&~~~i=n-2,\ldots,0.\nonumber
\end{align}


We claim that the cofactor is a constant, i.e., $k_1=k_2=k_3=0$. Indeed, under the change of (\ref{D8}),
we can transform (\ref{DD3}) into an ordinary differential equation if we fixe $u$ and $v$
\begin{equation*}\label{DD6}
-(\pm\sqrt{u-Aw^{2}})(\pm\sqrt{v-w^{2}})\frac{d\bar{f_n}}{dw}
=[k_1(\pm\sqrt{u-Aw^{2}})+k_2w+k_3(\pm\sqrt{v-w^{2}})]\bar{f}_n,
\end{equation*}
where $\bar{f}_n(u,v,w)=f_n(x,y,z)$.
In the following proof, we only consider the case of $xz<0$.
For the case of $xz>0$,
the proof is similar.
Solving the last equation we find that for $xz<0$,
\begin{align*}
\bar{f}_n=&\bar{A}(u,v)\left|\frac{2\sqrt{A}\sqrt{(u-Aw^2)(v-w^2)}+2Aw^2-(u+Av)}{2\sqrt{A}}
\right|^{-\frac{k_2}{2\sqrt{A}}}\\
&\exp\left(-k_1\arcsin\frac{w}{\sqrt{v}}\right)\exp\left(-\frac{k_3}{\sqrt{A}}\arcsin\frac{w}{\sqrt{u}}\right).
\end{align*}
In order for $f_n(x,y,z)=\bar{f_n}(u,v,w)$ to be a homogeneous polynomial of degree $n$ in $x,y,z$,
we have $k_1=k_3=0$ and the function $\bar{A}$ is a homogeneous polynomial in $x^2+Ay^2$ and $y^2+z^2$.
Then
\begin{eqnarray*}
f_n=\bar{A}(x^2+Ay^2,y^2+z^2)(x+\sqrt{A}z)^{\frac{k_2}{\sqrt{A}}}.
\end{eqnarray*}
Therefor $f$ is a Darboux polynomial of degree $n(=2m+\frac{k_2}{\sqrt{A}})$
with the cofactor $K=k_2y+k_0$.
Set $\bar{k}_2={k_2}/{\sqrt{A}}$, we get
$
f=\sum_{i=0}^{2m+\bar{k}_2}f_i
$
 and
\begin{eqnarray*}
f_{2m+\bar{k}_2}=(x+\sqrt{A}z)^{\bar{k}_2}\sum_{i=0}^{m}a_{i}^{m}(x^2+Ay^2)^{m-i}(y^2+z^2)^i.
\end{eqnarray*}
Introducing $f_{2m+\bar{k}_2}$ into (\ref{DD4}) and performing some calculations,
we obtain
\begin{align*}
Ayz&\frac{\partial f_{2m+\bar{k}_2-1}}{\partial x}-xz\frac{\partial f_{2m+\bar{k}_2-1}}{\partial y}+xy\frac{\partial f_{2m+\bar{k}_2-1}}{\partial z}-k_2yf_{2m+\bar{k}_2-1}\\
&=(x+\sqrt{A}z)^{\bar{k}_2}\sum_{i=0}^{m}a_{i}^{m}[2\sigma(m-i)+k_{0}+2i](x^2+Ay^2)^{m-i}(y^2+z^2)^i\\
&+(x+\sqrt{A}z)^{\bar{k}_2-1}\sum_{i=0}^{m}a_{i}^{m}\bar{k}_2\sigma(x^2+Ay^2)^{m-i}(y^2+z^2)^ix\\
&-(x+\sqrt{A}z)^{\bar{k}_2}\sum_{i=0}^{m}a_{i}^{m}2C(m-i)(x^2+Ay^2)^{m-i-1}(y^2+z^2)^ixz\\
&+(x+\sqrt{A}z)^{\bar{k}_2-1}\sum_{i=0}^{m}a_{i}^{m}\bar{k}_0(\sqrt{A}-C)(x^2+Ay^2)^{m-i}(y^2+z^2)^iz\\
&+(x+\sqrt{A}z)^{\bar{k}_2}\sum_{i=0}^{m}a_{i}^{m}2A(m-i)(1-\sigma)(x^2+Ay^2)^{m-i-1}(y^2+z^2)^iy^2.
\end{align*}
Using the transformations (\ref{D9}),
the above equation becomes
\begin{align}\label{DD7}
&\sqrt{u-Aw^2}\sqrt{v-w^2}\frac{d\bar{f}_{2m+\bar{k}_2-1}}{dw}-\bar{k}_2w\bar{f}_{2m+\bar{k}_2-1}\\\nonumber
&=(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})^{\bar{k}_2}\sum_{i=0}^{m}a_{i}^{m}[2\sigma(m-i)+k_{0}+2i]u^{m-i}v^i\\\nonumber
&+(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})^{\bar{k}_2-1}\sum_{i=0}^{m}a_{i}^{m}\bar{k}_2\sigma u^{m-i}v^i\sqrt{u-Aw^2}\\\nonumber
&+(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})^{\bar{k}_2}\sum_{i=0}^{m}a_{i}^{m}2C(m-i)u^{m-i-1}v^i\sqrt{u-Aw^2}\sqrt{v-w^2}\\\nonumber
&+(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})^{\bar{k}_2-1}\sum_{i=0}^{m}a_{i}^{m}\bar{k}_0(C-\sqrt{A})u^{m-i}v^i\sqrt{v-w^2}\\\nonumber
&+(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})^{\bar{k}_2}\sum_{i=0}^{m}a_{i}^{m}2A(m-i)(1-\sigma)u^{m-i-1}v^iw^2.
\end{align}
which is a non-homogeneous linear ordinary differential equation in $\bar{f}_{2m+\bar{k}_2-1}$,
The corresponding homogeneous equation
\begin{eqnarray*}
\sqrt{u-Aw^2}\sqrt{v-w^2}\frac{d\bar{f}^{*}_{2m+\bar{k}_2-1}}{dw}-\bar{k}_2w\bar{f}^{*}_{2m+\bar{k}_2-1}=0
\end{eqnarray*}
has a general solution
\begin{eqnarray*}
\bar{f}^{*}_{2m+\bar{k}_2-1}=(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})^{\bar{k}_2}\bar{A}^{*}_{2m-1}(u,v),
\end{eqnarray*}
where $\bar{A}^{*}_{2m-1}(u,v)$ is an arbitrary smooth function in $u$ and $v$.
In order to use the method of variation of constants,
we assume that
\begin{eqnarray*}
\bar{f}_{2m+\bar{k}_2-1}=(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})^{\bar{k}_2}\bar{A}_{2m-1}(u,v,w)
\end{eqnarray*}
is a solution of (\ref{DD7}),
then $\bar{A}_{2m-1}(u,v,w)$ satisfies
\begin{align*}
\frac{d\bar{A}_{2m-1}}{dw}&=\sum_{i=0}^{m}a_{i}^{m}[2\sigma(m-i)+b+2i]u^{m-i}v^i\frac{1}{\sqrt{u-Aw^2}\sqrt{v-w^2}}\\
&+\sum_{i=0}^{m}a_{i}^{m}\bar{k}_2\sigma u^{m-i}v^i\frac{1}{(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})\sqrt{v-w^2}}\\
&+\sum_{i=0}^{m}a_{i}^{m}2C(m-i)u^{m-i-1}v^i\\
&+\sum_{i=0}^{m}a_{i}^{m}\bar{k}_0(C-\sqrt{A})u^{m-i}v^i\frac{1}{(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})\sqrt{ u-Aw^2}}\\
&+\sum_{i=0}^{m}a_{i}^{m}2A(m-i)(1-\sigma)u^{m-i-1}v^i\frac{w^2}{\sqrt{u-Aw^2}\sqrt{v-w^2}}.
\end{align*}
Some easy computations lead to
\begin{eqnarray}\label{ss}
\int\frac{d w}{(\sqrt{u-Aw^2}-\sqrt{A}\sqrt{v-w^2})\sqrt{u-Aw^2}}
=\frac{w}{u-v}+\int\frac{\sqrt{v-w^2}}{\sqrt{u-Aw^2}}d w
\end{eqnarray}
Since
\begin{align*}
\int\frac{d w}{\sqrt{u-Aw^2}\sqrt{v-w^2}},~~~~~~~
\int\frac{\sqrt{u-Aw^2}}{\sqrt{v-w^2}}d w
\end{align*}
are elliptic integrals of the first and second kind, by (\ref{D15}) and (\ref{ss}),
in order for $A_{2m-1}(x,y,z)=\bar{A}_{2m-1}(u,v,w)$ to be a homogeneous polynomial
of degree $2m-1$, we must have
\begin{equation}\label{D26}
\begin{cases}
a_{i}^{m}[2\sigma(m-i)+k_0+2i]=0,\\
a_{i}^{m}\bar{k}_{2}\sigma=0,\\
a_{i}^{m}\bar{k}_{2}(C-\sqrt{A})=0,\\
a_{i}^{m}2A(m-i)(1-\sigma)=0, ~~~i=0,1,\cdots,m,
\end{cases}
\end{equation}
which implies $\bar{k}_2=0$, that is to say, $k_2=0$. So the cofactor $K=k_{0}$ is a constant.
Then (\ref{D26}) becomes
\begin{equation*}
\begin{cases}
a_{i}^{m}[2\sigma(m-i)+2i+k_{0}]=0,\\
a_{i}^{m}2A(m-i)(1-\sigma)=0, ~~~i=0,1,\cdots,m.
\end{cases}
\end{equation*}

Now, we split the proof in two cases.
\\
\emph{Case I}. $\sigma=1$, $k_{0}=-2m$. In this case, we have
\begin{align}
&f_{n}=f_{2m}=\sum_{i=0}^{m}a_{i}^{m}(x^2+Ay^2)^{m-i}(y^2+z^2)^{i},\label{ss1}\\
&f_{n-1}=f_{2m-1}=\sum_{i=0}^{m}a_{i}^{m}2C(m-i)(x^2+Ay^2)^{m-i-1}(y^2+z^2)^{i}y.\label{ss2}
\end{align}
Substituting (\ref{ss1}) and (\ref{ss2}) into (\ref{DD4}) with $i=n-2$, we get
\begin{align*}
Ayz\frac{\partial f_{n-2}}{\partial x}&-xz\frac{\partial f_{n-2}}{\partial y}+xy\frac{\partial f_{n-2}}{\partial z}\\
=&-\sum_{i=0}^{m}a_{i}^{m}4C^{2}(m-i)(m-i-1)(x^2+Ay^2)^{m-i-2}(y^2+z^2)^ixyz\\
&-\sum_{i=0}^{m}a_{i}^{m}2Rai(x^2+Ay^2)^{m-i}(y^2+z^2)^{i-1}z\\
&-\sum_{i=0}^{m}a_{i}^{m}4AC(m-i)(x^2+Ay^2)^{m-i-2}(y^2+z^2)^{i}y^{3}.
\end{align*}
By (\ref{D9}),
the above equation becomes
\begin{align*}
\frac{d{\bar{f}}_{n-2}}{dw}=&\sum_{i=0}^{m}a_{i}^{m}4C^{2}(m-i)(m-i-1)u^{m-i-2}v^iw\\
&+\sum_{i=0}^{m}a_{i}^{m}2Raiu^{m-i}v^{i-1}\frac{1}{\sqrt{u-Aw^{2}}}\\
&-\sum_{i=0}^{m}a_{i}^{m}2C(m-i)u^{m-i-1}v^{i}\frac{w}{\sqrt{u-Aw^{2}}\sqrt{v-w^{2}}},
\end{align*}
that is,
\begin{align*}
\bar{f}_{n-2}&=\sum_{i=0}^{m}a_{i}^{m}2C^{2}(m-i)(m-i-1)u^{m-i-2}v^i{w^2}\\
&+\sum_{i=0}^{m}a_{i}^{m}2Raiu^{m-i}v^{i-1}\frac{1}{\sqrt{A}}\arctan\left(\frac{\sqrt{A}w}{\sqrt{u-Aw^2}}\right)\\
&-\sum_{i=0}^{m}a_{i}^{m}2C(m-i)u^{m-i-1}v^{i}\frac{\ln|\sqrt{u-Aw^2}+\sqrt{A(v-w^2)}|}{\sqrt{A}}.
\end{align*}
We must have
\begin{equation*}
\begin{cases}
a_{i}^{m}2i Ra=0,\\
a_{i}^{m}2C(m-i)=0,
 ~~~i=0,1,\cdots,m,
\end{cases}
\end{equation*}
which yields $
a_{i}^{m}=0$ for $i=0,1,\cdots,m.
$
This is a contradiction to $f_n\neq0$.
\\
\\
\emph{Case II}. $\sigma\neq1$, $a_{i}^{m}=0$, $i=0,1,\cdots,m-1$, $k_{0}=-2\sigma m$. In this case, we have
\begin{align*}
f_{n}=f_{2m}=a_{m}^{m}(y^2+z^2)^{m},~~f_{n-1}=f_{2m-1}=0.
\end{align*}
Working in a
similar way to the previous case, we get
\begin{align*}
-\sqrt{u-Aw^{2}}\sqrt{v-w^{2}}\frac{d\bar{f}_{n-2}}{dw}
=-2a_{m}^{m}mRav^{m-1}\sqrt{v-w^{2}},
\end{align*}
and
\begin{align*}
\bar{f}_{n-2}
=2a_{m}^{m}mRav^{m-1}\frac{1}{\sqrt{A}}\arctan\left(\frac{\sqrt{A}w}{\sqrt{u-Aw^2}}\right).
\end{align*}
Therefore, $a_{m}^{m}=0$  and $f_n=0$, which is a contradiction.


\begin{proposition}{Proposition}\label{T3}
System (\ref{I1}) has no exponential factors.
\end{proposition}
{\bf Proof:}
Let $E = \exp(g/h)$ be an
exponential factor of system (\ref{I1}) with a cofactor
$$
L=l_0+l_1x+l_2y+l_3z,~~l_i\in\mathbb{R},~~i=0,1,2,3,
$$
 where $g,h\in\mathbb{R}[x,y,z]$ with
$f,g$ being prime. From Proposition \ref{PP2}, \ref{T1} and \ref{T2},  $E = \exp(g)$ with $g=g(x,y,z)\in \mathbb{R}[x,y,z]/{\mathbb{R}}$.
By definitions,  $g$ satisfies
\begin{align}\label{E1}
(Ayz+Cz-\sigma x)\frac{\partial g}{\partial x}+(-xz+Ra-y)\frac{\partial g}{\partial y}&+(-z+xy)\frac{\partial g}{\partial z}=l_{0}+l_{1}x+l_{2}y+l_{3}z.
\end{align}

If  $g$ is a polynomial of degree $n\geq3$.
We write $g$ as $g=\sum_{j=0}^{n}g_{j}(x,y,z)$,
where each $g_{j}$ is a homogeneous polynomial of degree $j$ and $g_n\neq 0$.
Observing the right hand side of (\ref{E1}) has degree at most one,
we compute the terms of degree $n+1$ in (\ref{E1}) and get
\begin{equation*}\label{E2}
Ayz\frac{\partial g_{n}}{\partial x}-xz\frac{\partial g_{n}}{\partial y}+xy\frac{\partial g_{n}}{\partial z}=0,
\end{equation*}
which is (\ref{po}) replacing $f_n$ by $g_n$. Then the arguments used in the proof of Proposition \ref{T1} imply
$g_{n}=L_{n}(y^{2}+z^{2})^{m}$ with $L_{n}\in\mathbb{R}$.
Now computing the terms of degree $n$ in (\ref{E1}) leads to
\begin{align*}\label{E3}
Ayz\frac{\partial g_{n-1}}{\partial x}-xz\frac{\partial g_{n-1}}{\partial y}+xy\frac{\partial g_{n-1}}{\partial z}&=(\sigma x-Cz)\frac{\partial g_{n}}{\partial x}+y\frac{\partial g_{n}}{\partial y}+z\frac{\partial g_{n}}{\partial z},
\end{align*}
which is  (\ref{assd}) with $f_n$ replaced by $g_n$ and $f_{n-1}$ replaced by $g_{n-1}$. Again, the arguments used in the
proof of Proposition \ref{T1} imply that $g_n=0$, which is a contradiction.


Hence, $g$ is a polynomial of degree at most two satisfying (\ref{E1}).
So we can write $g$ as
\begin{equation}\label{E4}
g=b_{0}+b_{1}x+b_{2}y+b_{3}z+b_{11}x^{2}+b_{22}y^{2}+b_{33}z^{2}+b_{12}xy+b_{13}xz+b_{23}yz.
\end{equation}
Then substituting (\ref{E4}) into (\ref{E1}) yields a system of  algebraic equations
 $g_i=0$  for $i=1,\cdots,15$, where
\begin{align*}
&g_1=Ra b_{2}-l_0,~g_2=-\sigma b_{1}+Ra b_{12}-l_1,\\
&g_3=2Ra b_{22}- b_{2}-l_2,~g_4=Cb_{1}+Ra b_{23}-b_3-l_3,\\
&g_5=-\sigma b_{12}-b_{12}+b_{3},~g_6=-\sigma b_{13}-b_{2}-b_{13},\\
&g_7=A b_{1}+Cb_{12}-2b_{23},~g_8=-2\sigma b_{11},\\
&g_9=-2b_{22},~g_{10}=2Cb_{11}+Cb_{13}-2b_{33},~g_{11}=-b_{22}+b_{33},\\
&g_{12}=b_{13},
~g_{13}=b_{12},
~g_{14}=b_{23},
~g_{15}=2Ab_{11}+Ab_{13}.
\end{align*}
From the above equations,
we obtain that $l_{0}=l_{1}=l_{2}=l_{3}=0$, $b_{1}=b_{2}=b_{3}=0$, $b_{11}=b_{22}=b_{33}=b_{12}=b_{13}=b_{23}=0$,
which implies $g$ is a constant.
This completes the proof.


\noindent \emph{ Proof of Theorem \ref{th}}: By Proposition \ref{PP1} and \ref{T1}-\ref{T3}, it is obvious.





\section{Appendix}

In this section, we provide a relationship between the Glukhovsky-Dolzhansky system (\ref{I1}) and other similar quadric
systems including the Rabinovich system,  the 3D forced-damped system and the \emph{D2} vector field.


 The  Rabinovich system \cite{si1,si2} is a three-wave interaction model and is given by
 \begin{equation}\label{rr}
\begin{cases}
\dot{X}=h Y-v_{1} X+YZ \\
\dot{Y}=h X-v_{2} Y-XZ \\
\dot{Z}=-v_{3} Z+XY.
\end{cases}
\end{equation}
where $h,~v_1,~v_2,~v_3$ are parameters.  When $C=-2ARa$, by a family of linear scaling $\mathcal{S}_{\alpha}: (x,y,z,t)\rightarrow (X,Y,Z,T)$
$$
x=\frac{X}{\alpha},~~y=Ra-\frac{Z}{\alpha\sqrt{-A}},~~z=\frac{Y}{\alpha\sqrt{-A}},~~t=\alpha T,
$$
we transform the Glukhovsky-Dolzhansky system (\ref{I1}) into
 \begin{equation}\label{rr2}
\begin{cases}
\frac{dX}{dT}=\alpha\sqrt{-A}Ra Y-\alpha\sigma X+YZ \\
\frac{dY}{dT}=\alpha\sqrt{-A}Ra X-\alpha Y-XZ \\
\frac{dZ}{dT}=-\alpha Z+XY.
\end{cases}
\end{equation}
Clearly, system (\ref{rr2}) corresponds to the Rabinovich system (\ref{rr}) with parameters $(h,v_1,v_2,v_3)=(\alpha\sqrt{-A}Ra, \alpha\sigma,\alpha,\alpha)$. Hence, if $C=-2ARa$, the Glukhovsky-Dolzhansky system (\ref{I1})
is equivalent to the Rabinovich system (\ref{rr}) in the particular case of the parameter region $\{(h,v_1.v_2.v_3)|v_2=v_3\}$, that is,  two systems are homothetic copies. In addition, the linear  scaling
$\mathcal{S}_{\alpha}$ forces parameter $A$ is less than zero, which cannot hold in the physical region of parameter $A$.


The 3D forced-damped system \cite{si3} arises in mechanical, electrical and fluid dynamical contexts and is given by
 \begin{equation}\label{3ff}
\begin{cases}
\dot{X}=-aX+Y+YZ \\
\dot{Y}=X-aY+bXZ \\
\dot{Z}=cZ-bXY
\end{cases}
\end{equation}
where $a,b,c$ are parameters. Compared with  the Glukhovsky-Dolzhansky system (\ref{I1}), the 3D forced-damped system (\ref{3ff})
has less parameters. When $\sigma=1$, the linear scaling
$$
x=\frac{(ARa+C)^{3/2}}{A\sqrt{Ra}}X,~~y=Ra+\frac{(ARa+C)Z}{A},~~z=\frac{(ARa+C)Y}{A},~~t=\frac{T}{\sqrt{(ARa+C)Ra}},
$$
transforms the Glukhovsky-Dolzhansky system (\ref{I1}) into
 \begin{equation}\label{3ff2}
\begin{cases}
\frac{dX}{dT}=-\frac{1}{\sqrt{(ARa+C)Ra}}X+Y+YZ \\
\frac{dY}{dT}=X-\frac{1}{\sqrt{(ARa+C)Ra}}Y+\frac{ARa+C}{{ARa}}XZ \\
\frac{dZ}{dT}=-\frac{1}{\sqrt{(ARa+C)Ra}}Z-\frac{ARa+C}{{ARa}}XY,
\end{cases}
\end{equation}
which corresponds to the 3D forced-damped system  (\ref{3ff}) with parameters
$$
(a,b,c)=(\frac{1}{\sqrt{(ARa+C)Ra}}, \frac{ARa+C}{{ARa}},-\frac{1}{\sqrt{(ARa+C)Ra}}).
$$
Therefore, when $\sigma=1$, the Glukhovsky-Dolzhansky system (\ref{I1})
is equivalent to the 3D forced-damped system (\ref{3ff}) in the particular case of the parameter region $\{(a,b,c)|a+c=0\}$.


Consider the vector field equivariant under the \emph{D2} symmetry
group, called the \emph{D2} vector field \cite{si4}
 \begin{equation}\label{DD2}
\begin{cases}
\dot{X}=aX+YZ \\
\dot{Y}=bY+XZ \\
\dot{Z}=Z-XY.
\end{cases}
\end{equation}
where $a,b$ are parameters. Similar to above, when $Ra=C=0$, we make a linear scaling
$$
x=X,~~y=\frac{Z}{\sqrt{A}},~~z=-\frac{Y}{\sqrt{A}},~~t=-T
$$
and transform (\ref{I1}) into
 \begin{equation}\label{DD22}
\begin{cases}
\frac{dX}{dT}=\sigma X+YZ \\
\frac{dY}{dT}=Y+XZ \\
\frac{dZ}{dT}=Z-XY.
\end{cases}
\end{equation}
which corresponds to the \emph{D2} vector field (\ref{DD2})   with parameters $(a,b)=(\sigma,1)$.
Then, when $Ra=C=0$, the Glukhovsky-Dolzhansky system (\ref{I1})
is equivalent to the \emph{D2} vector field (\ref{DD2}) in the particular case of the parameter region $\{(a,b)|b=1\}$.





\begin{thebibliography}{99}


\bibitem{model}
A.B. Glukhovsky,  F.A. Dolzhansky, Three-component geostrophic models of convection in a rotating fluid, \emph{Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana} {\bf 16}(1980) 311-318.


\bibitem{gb0}
G.A. Leonov, N.V. Kuznetsov, T.N. Mokaev,
Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity,
\emph{Commun. Nonlinear Sci. Numer. Simul.} {\bf 28}(2015) 166-174.

\bibitem{gb1}
G. Chen, N.V. Kuznetsov, G.A. Leonov, N.V. Kuznetsov, Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems,
\emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.} {\bf27}(2017) 1750115, 9 pp.

\bibitem{gb2}
G.A. Leonov, T.N. Mokaev, Lyapunov dimension formula for the attractor of the Glukhovsky-Dolzhansky system,
 \emph{Doklady Mathematics} {\bf93}(2016) 42-45.


\bibitem{gb3}
G.A. Leonov, Existence criterion of homoclinic trajectories in the Glukhovsky-Dolzhansky system, \emph{Phys. Lett. A}
{\bf379}(2015) 524-528.


\bibitem{in1}
X. Zhang, Integrability of dynamical systems: algebra and analysis, Springer-Verlag, 2017.


\bibitem{in3}
K. Huang, S. Shi and W. Li, Kovalevskaya exponents, weak Painlev\'{e} property and integrability for quasi-homogeneous differential systems,  \emph{Regul. Chaotic Dyn.} {\bf25}(2020) 295-312.












\bibitem{dd2}
J. Llibre, Y. Paulina and C. Valls, On the global dynamics of a three-dimensional forced-damped differential system,
 \emph{ J. Nonlinear
Math. Phys.} {\bf27}(2020) 414-428.


\bibitem{dd3}
K. Huang, S. Shi and Z. Xu, Integrable deformations, bi-Hamiltonian structures and nonintegrability of a generalized Rikitake system,
\emph{Int. J. Geom. Methods Mod. Phys.} {\bf16}(2019) 1950059, 17 pp.

\bibitem{dar1}
F. Dumortier, J. Llibre and J.C. Art\'{e}s,  Qualitative Theory of Planar Differential Systems, in: Universitext, Springer-Verlag, Berlin, 2006.


\bibitem{dar2}
J. Llibre, X. Zhang, Darboux theory of integrability in Cn taking into account the multiplicity, \emph{J. Differential Equations}
{\bf246}(2009) 541-551.



\bibitem{dar3}
C. Christopher, J. Llibre, Integrability via invariant algebraic curves for planar polynomial
differential systems, \emph{Ann. Differential Equations} {\bf16}(2000) 5-19.


\bibitem{dar4}
C. Christopher, J. Llibre, J.V. Pereira,  Multiplicity of invariant algebraic curves in polynomial vector fields,
\emph{Pacific J. Math.} {\bf229}(2007) 63-117.



\bibitem{dar5}
J. Llibre, X. Zhang, On the Darboux Integrability of Polynomial Differential Systems, \emph{Qual. Theory Dyn. Syst.}
{\bf 11}(2012) 129-144.

\bibitem{d1}
J. Llibre, C. Valls, On the integrability of the 5-dimensional Lorenz system for the gravity-wave activity,
\emph{ Proc. Amer. Math. Soc.} {\bf145}(2017) 665-679.


\bibitem{d2}
C. Valls,
Invariant algebraic surfaces for generalized Raychaudhuri equations.
\emph{Comm. Math. Phys.} {\bf308}(2011) 133-146.


\bibitem{d3}
C. Valls, Invariant algebraic surfaces for a virus dynamics, \emph{Z. Angew. Math. Phys.} {\bf 66}(2015) 1315-1328.


\bibitem{d4}
C. Valls, Invariant algebraic surfaces and algebraic first integrals of the Maxwell-Bloch system,\emph{J. Geom. Phys.}
{\bf 146}(2019) 103516, 8 pp.

\bibitem{d5}
M.R. Candido, J. Llibre, C. Valls, Invariant algebraic surfaces and Hopf bifurcation of a finance model,
\emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.} {\bf28}(2018) 1850150, 10 pp.

\bibitem{dd1}
C. Valls, On the global dynamics of the Newell-Whitehead system,
\emph{J. Nonlinear Math. Phys.} {\bf 26}(2019)  569-578.




\bibitem{ga1}
M. Ayoul, N.T. Zung, Galoisian obstructions to non-Hamiltonian integrability,
\emph{ C. R. Math. Acad. Sci. Paris} {\bf348}(2010) 1323-1326.

\bibitem{ga2}
K. Huang, S. Shi and W. Li, Meromorphic non-integrability of several 3D dynamical systems,
\emph{Entropy} {\bf19}(2017) 211, 16 pp.


\bibitem{ga3}
W. Szuminski, M. Przybylska, Differential Galois integrability obstructions for nonlinear three-dimensional differential systems,
 \emph{Chaos} {\bf 30}(2020) 013135, 12 pp.





\bibitem{gaa1}
K. Huang, S. Shi and W. Li, Integrability analysis of the Shimizu-Morioka system.
 \emph{Commun. Nonlinear Sci. Numer. Simul.} {\bf84}(2020) 105101, 12 pp.

\bibitem{gaa2}
K. Huang, S. Shi and W. Li,  Meromorphic and formal first integrals for the Lorenz system,  \emph{J. Nonlinear Math. Phys.}
 {\bf25}(2018), 106-121.

\bibitem{gaa3}
K. Huang, S. Shi and W. Li,
First integrals of the Maxwell-Bloch system, \emph{ C. R. Math. Acad. Sci. Paris} {\bf358}(2020) 3-11.


\bibitem{newgaa3}
A.J. Maciejewskia and M. Przybylska, Integrability analysis of the stretch-twist-fold flow, \emph{J. Nonlinear Sci.} {\bf30}(2020),  1607-1649.


\bibitem{si1}
X. Zhang,
Integrals of motion of the Rabinovich system, \emph{J. Phys. A} {\bf33}(2000)  5137-5155.
\bibitem{si2}
F. Xie and X. Zhang, Invariant algebraic surfaces of the Rabinovich system, \emph{J. Phys. A}  {\bf36}(2003) 499-516.
\bibitem{si3}
J. Llibre, R. Oliveira,  C. Valls,
On the Darboux integrability of a three-dimensional forced-damped differential system,
\emph{J. Nonlinear Math. Phys.} {\bf24}(2017), 473-494.
\bibitem{si4}

K. Katsios, S. Anastassiou,
Darboux polynomials and global phase portraits for the D2
vector field,
\emph{J. Math. Anal. Appl.} {\bf475}(2019) 32-40.

\end{thebibliography}

\end{document}
2102.06549.extracted_bib
{model A.B. Glukhovsky, F.A. Dolzhansky, Three-component geostrophic models of convection in a rotating fluid, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana {16(1980) 311-318.
2102.06549.extracted_bib
{gb0 G.A. Leonov, N.V. Kuznetsov, T.N. Mokaev, Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity, Commun. Nonlinear Sci. Numer. Simul. {28(2015) 166-174.
2102.06549.extracted_bib
{gb1 G. Chen, N.V. Kuznetsov, G.A. Leonov, N.V. Kuznetsov, Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. {27(2017) 1750115, 9 pp.
2102.06549.extracted_bib
{gb2 G.A. Leonov, T.N. Mokaev, Lyapunov dimension formula for the attractor of the Glukhovsky-Dolzhansky system, Doklady Mathematics {93(2016) 42-45.
2102.06549.extracted_bib
{gb3 G.A. Leonov, Existence criterion of homoclinic trajectories in the Glukhovsky-Dolzhansky system, Phys. Lett. A {379(2015) 524-528.
2102.06549.extracted_bib
{in1 X. Zhang, Integrability of dynamical systems: algebra and analysis, Springer-Verlag, 2017.
2102.06549.extracted_bib
{in3 K. Huang, S. Shi and W. Li, Kovalevskaya exponents, weak Painlev\'{e property and integrability for quasi-homogeneous differential systems, Regul. Chaotic Dyn. {25(2020) 295-312.
2102.06549.extracted_bib
{dd2 J. Llibre, Y. Paulina and C. Valls, On the global dynamics of a three-dimensional forced-damped differential system, J. Nonlinear Math. Phys. {27(2020) 414-428.
2102.06549.extracted_bib
{dd3 K. Huang, S. Shi and Z. Xu, Integrable deformations, bi-Hamiltonian structures and nonintegrability of a generalized Rikitake system, Int. J. Geom. Methods Mod. Phys. {16(2019) 1950059, 17 pp.
2102.06549.extracted_bib
{dar1 F. Dumortier, J. Llibre and J.C. Art\'{es, Qualitative Theory of Planar Differential Systems, in: Universitext, Springer-Verlag, Berlin, 2006.
2102.06549.extracted_bib
{dar2 J. Llibre, X. Zhang, Darboux theory of integrability in Cn taking into account the multiplicity, J. Differential Equations {246(2009) 541-551.
2102.06549.extracted_bib
{dar3 C. Christopher, J. Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential Equations {16(2000) 5-19.
2102.06549.extracted_bib
{dar4 C. Christopher, J. Llibre, J.V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math. {229(2007) 63-117.
2102.06549.extracted_bib
{dar5 J. Llibre, X. Zhang, On the Darboux Integrability of Polynomial Differential Systems, Qual. Theory Dyn. Syst. {11(2012) 129-144.
2102.06549.extracted_bib
{d1 J. Llibre, C. Valls, On the integrability of the 5-dimensional Lorenz system for the gravity-wave activity, Proc. Amer. Math. Soc. {145(2017) 665-679.
2102.06549.extracted_bib
{d2 C. Valls, Invariant algebraic surfaces for generalized Raychaudhuri equations. Comm. Math. Phys. {308(2011) 133-146.
2102.06549.extracted_bib
{d3 C. Valls, Invariant algebraic surfaces for a virus dynamics, Z. Angew. Math. Phys. {66(2015) 1315-1328.
2102.06549.extracted_bib
{d4 C. Valls, Invariant algebraic surfaces and algebraic first integrals of the Maxwell-Bloch system,J. Geom. Phys. {146(2019) 103516, 8 pp.
2102.06549.extracted_bib
{d5 M.R. Candido, J. Llibre, C. Valls, Invariant algebraic surfaces and Hopf bifurcation of a finance model, Internat. J. Bifur. Chaos Appl. Sci. Engrg. {28(2018) 1850150, 10 pp.
2102.06549.extracted_bib
{dd1 C. Valls, On the global dynamics of the Newell-Whitehead system, J. Nonlinear Math. Phys. {26(2019) 569-578.
2102.06549.extracted_bib
{ga1 M. Ayoul, N.T. Zung, Galoisian obstructions to non-Hamiltonian integrability, C. R. Math. Acad. Sci. Paris {348(2010) 1323-1326.
2102.06549.extracted_bib
{ga2 K. Huang, S. Shi and W. Li, Meromorphic non-integrability of several 3D dynamical systems, Entropy {19(2017) 211, 16 pp.
2102.06549.extracted_bib
{ga3 W. Szuminski, M. Przybylska, Differential Galois integrability obstructions for nonlinear three-dimensional differential systems, Chaos {30(2020) 013135, 12 pp.
2102.06549.extracted_bib
{gaa1 K. Huang, S. Shi and W. Li, Integrability analysis of the Shimizu-Morioka system. Commun. Nonlinear Sci. Numer. Simul. {84(2020) 105101, 12 pp.
2102.06549.extracted_bib
{gaa2 K. Huang, S. Shi and W. Li, Meromorphic and formal first integrals for the Lorenz system, J. Nonlinear Math. Phys. {25(2018), 106-121.
2102.06549.extracted_bib
{gaa3 K. Huang, S. Shi and W. Li, First integrals of the Maxwell-Bloch system, C. R. Math. Acad. Sci. Paris {358(2020) 3-11.
2102.06549.extracted_bib
{newgaa3 A.J. Maciejewskia and M. Przybylska, Integrability analysis of the stretch-twist-fold flow, J. Nonlinear Sci. {30(2020), 1607-1649.
2102.06549.extracted_bib
{si1 X. Zhang, Integrals of motion of the Rabinovich system, J. Phys. A {33(2000) 5137-5155.
2102.06549.extracted_bib
{si2 F. Xie and X. Zhang, Invariant algebraic surfaces of the Rabinovich system, J. Phys. A {36(2003) 499-516.
2102.06549.extracted_bib
{si3 J. Llibre, R. Oliveira, C. Valls, On the Darboux integrability of a three-dimensional forced-damped differential system, J. Nonlinear Math. Phys. {24(2017), 473-494.
2102.06549.extracted_bib
{si4 K. Katsios, S. Anastassiou, Darboux polynomials and global phase portraits for the D2 vector field, J. Math. Anal. Appl. {475(2019) 32-40.