\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 274, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/274\hfil  Nonexistence of first integral]
{Higher order criterion for the nonexistence of formal first
integral for nonlinear systems}

\author[Z. G. Xu, W. L. Li, S. Y. Shi \hfil EJDE-2017/274\hfilneg]
{Zhiguo Xu, Wenlei Li, Shaoyun Shi}

\address{Zhiguo Xu \newline
School of Mathematics,
Jilin University,
Changchun 130012, China}
 \email{xuzg2014@jlu.edu.cn}

\address{Wenlei Li (corresponding author) \newline
School of Mathematics,
Jilin University,
Changchun 130012, China}
\email{lwlei@jlu.edu.cn, phone +86-431-85166042}

\address{Shaoyun Shi \newline
School of Mathematics
\& State key laboratory of automotive simulation and control,
Changchun 130012, China}
\email{shisy@jlu.edu.cn}

\dedicatory{Communicated by Zhaosheng Feng}

\thanks{Submitted April 13 2016. Published November 5, 2017.}
\subjclass[2010]{34C14, 34M25}
\keywords{Normal form theory; first integral; resonance}

\begin{abstract}
 The main purpose of this article is to find a criterion for the nonexistence
 of formal first integrals for nonlinear systems under general resonance.
 An algorithm illustrates an application to a class of generalized
 Lokta-Volterra systems. Our result generalize the classical Poincar\'e's
 nonintegrability theorem and the existing results in the literature.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}\label{sec1}

Let us consider the system
\begin{equation}\label{xaf0}
\dot {\mathbf{x}}=f(\mathbf{x}), \quad \mathbf{x}=(x_1,\dots ,x_n)^T\in\mathbb C^n,
\end{equation}
where $f(\mathbf{x})$ is an $n$-dimensional vector-valued
analytic function with $f(\mathbf{0})=\mathbf{0}$.

In 1891, Poincar\'e \cite{Poincare} provided a criterion on the nonexistence
of analytic or formal first integrals for system \eqref{xaf0}.

\begin{theorem}[Poincar\'e's nonintegrability theorem]\label{poincare}
If the eigenvalues $\lambda_{1},\dots ,\lambda_{n}$ of the Jacobian matrix
$A=Df(\mathbf{0})$ are $\mathbb{N}$-non-resonant, i.e., they do not
satisfy any resonant relations of the form
\[
\sum^{n}_{j=1}{k_j}\lambda_{j}=0,\quad
 k_{j}\in\mathbb{Z}_+ ,\quad \sum^{n}_{j=1}k_{j}\geq1,
\]
where $\mathbb{Z}_+=\mathbb{N}\cup\{0\}$, then system \eqref{xaf0}
has neither analytic nor formal first integrals in any neighborhood of the origin
$\mathbf{x}=\mathbf{0}$.
\end{theorem}

In 1996, Furta \cite{Furta} gave another proof to Theorem \ref{poincare}
with the additional assumption that the matrix $A$ can be diagonalized,
and furthermore, he obtained a result about the nonexistence of formal
integral for semi-quasi-homogeneous systems.
In 2001, Shi and Li \cite{Shi2001} presented a different proof to
Theorem \ref{poincare} without the assumption of diagonalization of $A$.
In 2007, Shi \cite{Shi2007} extended Theorem \ref{poincare} and provided
a necessary condition for system \eqref{xaf0} to have a rational first integral.
Cong et al \cite{Cong2011} in 2011 generalized the Shi's result to a
more general case. For related information, see \cite{Goriely} and
references therein.

We note that the above results are all obtained under the condition that
the eigenvalues of $A$ are non-resonant. How about the case when the
eigenvalues of $A$ are resonant? In 2003, Li et al \cite{Liwg} studied
the case that $A$ has a zero eigenvalue and others are non-resonant.
In 2010, Liu et al \cite{Liu} studied the case that several eigenvalues of
$A$ are zero and the others are non-resonant, and Li et al \cite{Liwl}
studied the case that the eigenvalues of $A$ are pairwise resonant.
In this paper, we will extend the above results to the more general case,
i.e., the eigenvalues of $A$ are in general resonance. With the help of
the theory of normal form \cite{Arnold} and power transformations \cite{Bruno89},
we give a criterion on the nonexistence of formal first integral for
system \eqref{xaf0} under the case that the eigenvalues of $A$ are in general
resonance.

This paper is organized as follows. In section 2, we will build a criterion
of nonexistence of formal first integral of nonlinear systems under general
 resonance. In section 3, an algorithm to determine the nonexistence of
formal first integrals of nonlinear systems
will be illustrated by studying a class of generalized Lotka-Volterra systems.


\section{Main results}
System \eqref{xaf0} can be rewritten as
\begin{equation}\label{xaf}
\dot {\mathbf{x}}=A\mathbf{x}+F(\mathbf{x})
\end{equation}
near some neighborhood of $\mathbf{x}=\mathbf{0}$, where $A=Df(\mathbf{0})$,
$F(\mathbf{x})=o(\|\mathbf{x}\|)$.
Let $\Lambda=(\lambda_1,\dots ,\lambda_n)^T$, and
\begin{equation*}
\Theta=\big\{\mathbf{k}=(k_1,\dots ,k_n)^T\in \mathbb
Z^n : \langle \mathbf{k},\Lambda\rangle=\sum_{i=1}^nk_i\lambda_i=0\big\},
\end{equation*}
then $\Theta$ is a subgroup of $\mathbb{Z}^n$. Let rank $\Theta$ denote
the number of the least generating elements of $\Theta$.

We assume that $A$ is a diagonalizable matrix, and without loss of generality,
$A=\operatorname{diag}(\lambda_1,\dots ,\lambda_n)$. By the Poincar\'{e}-Dulac
normal form theory \cite{Arnold}, there exists a formal transformation
$\mathbf{x}=\mathbf{y}+h(\mathbf{y})$, such that system \eqref{xaf} can be changed to
\begin{equation}\label{2.7}
\dot y_i=\lambda_iy_i+\tilde F^i(\mathbf{y}),\quad i=1,\dots ,n,
\end{equation}
where $\mathbf{y}=(y_1,\dots ,y_n)^T$,
\begin{equation}\label{tildeF}
\tilde F^i(\mathbf{y})=\sum_{r=2}^{\infty}\sum_{|\mathbf{k}|=r,\langle
\mathbf{k},\Lambda\rangle=\lambda_i}\tilde F^i_{\mathbf{k}}\mathbf{y}^{\mathbf{k}},
\quad i=1,\dots , n,
\end{equation}
here $|{\mathbf{k}}|=\sum\limits_{i=1}^mk_i$,
$\mathbf{y}^{\mathbf{k}}=y_1^{k_1}\dots y_n^{k_n}, \tilde F^i_{\mathbf{k}}$
are constant coefficients.
For convenience, we give the following definition.

\begin{definition}\label{defresonant} \rm
A function $\Phi(\textbf{x})$ is resonant with respect to $\Lambda$,
if it can be written as
\begin{equation*}
\Phi(\mathbf{x})=\sum_{i=1}^{\infty}\sum_{|\mathbf{k}|=i, \langle
\mathbf{k},\Lambda\rangle=0}\Phi_{\mathbf{k}}{\mathbf{x}}^{\mathbf{k}},
\end{equation*}
where $\Phi_{\mathbf{k}}$ are constants.
\end{definition}

\begin{lemma}[\cite{Liwl,Walcher}]\label{l3}
If system \eqref{xaf} admits a nontrivial formal first integral
$\Phi(\mathbf{x})$, then system \eqref{2.7} has a nontrivial formal
first integral $\tilde\Phi(\mathbf{y})$ which is resonant with respect
to $\Lambda$.
\end{lemma}

\begin{lemma}\label{theta}
If $\operatorname{rank} \Theta=m$ $(0<m<n)$, then there exist $n-m$ eigenvalues which
are $\mathbb N$-non-resonant.
\end{lemma}

\begin{proof}
Assume that any $n-m$ eigenvalues of matrix $A$ are $\mathbb N$-resonant,
then there exist $n-m$ constants $a^1_{1},\dots ,a^1_{n-m}\in\mathbb Z_+$
with $a^1_{1}+\dots +a^1_{n-m}\ne 0$ such
that
$$
a^1_{1}\lambda_1+\dots +a^1_{n-m}\lambda_{n-m}=0.
$$
Thus $\mathbf{\tau_1}=(a^1_{1},\dots ,a^1_{n-m},0,\dots ,0)^T\in\Theta$. Without
loss of generality, we assume that $a^1_{1}\neq 0$.

Analogously, there exist $n-m$ constants
$a^2_{1},\dots ,a^2_{n-m}\in\mathbb Z_+$ with
$a^2_{1}+\dots +a^2_{n-m}\ne 0$ such that
$$
a^2_{2}\lambda_2+\dots +a^2_{n-m+1}\lambda_{n-m+1}=0.
$$
Thus $\mathbf{\tau_2}=(0,a^2_{2},\dots ,a^2_{n-m+1},0,\dots ,0)^T\in\Theta$.
Similarly, we assume that $a^2_{2}\neq 0$.

Repeating the above process, we obtain
$$
\mathbf{\tau_i}=(0,\dots ,0,a^i_{i},\dots ,
a^i_{n-m+i-1},0\dots ,0)^T\in\Theta, ~~a^i_{i}\neq0,~~i=1,\dots ,m+1.
$$
Obviously, $\mathbf{\tau_1},\dots ,\mathbf{\tau_{m+1}}$ are linear
independent. This is contradict to the fact that rank $\Theta=m$,
the proof is complete.
\end{proof}

For simplicity of presentation, we make the following assumption.
\begin{itemize}
\item[(H1)] $A=\operatorname{diag}(\lambda_1,\dots ,\lambda_n)$,
$\operatorname{rank}\Theta=m$ $(0<m<n)$, and $\lambda_{m+1},\dots ,\lambda_{n}$
are $\mathbb N$-non-resonant.
Let
\begin{equation*}
\Theta'=\big\{\mathbf{k}\in \Theta: \sum_{i=1}^nk_{i}\geq 1,
 \mbox{ and there exists $j\in\{1,\dots ,n\}$ such that }
\mathbf{k}+\mathbf{e}_j\geq 0\big\},
\end{equation*}
here $\mathbf{e}_1,\dots ,\mathbf{e}_n$ are the natural bases of
$\mathbb R^n$, and a vector $\mathbf a\geq 0$ means every component
$a_i \geq 0$, for $i=1,\dots,n$.
\end{itemize}

\begin{lemma}\label{lem1}
Assume {\rm (H1)} holds, then there exist $m$ linear independent
vectors $\mathbf{\tau_1},\dots ,\mathbf{\tau_m}\in\mathbb Q^n$, such
that
\begin{itemize}
\item[(1)] for any $\mathbf{k}\in \Theta$, there exists
$\mathbf a=(a_1,\dots , a_m)^T\in\mathbb Q^m$, such that
$\mathbf{k}=a_1\mathbf{\tau_1}+\dots +a_m\mathbf{\tau_m}$;

\item[(2)] for any $\mathbf{k}\in \Theta'$, there exists $\mathbf
a=(a_1,\dots , a_m)^T\in\mathbb Z^m_+$, $|\mathbf
a|\geq 1$, such that
$\mathbf{k}=a_1\mathbf{\tau_1}+\dots +a_m\mathbf{\tau_m}$.
\end{itemize}
\end{lemma}

\begin{proof}
Let $B=(\mathbf{\alpha_1},\dots ,\mathbf{\alpha_m})^T$,
$P_1=(\mathbf{\alpha_{m+1}},\dots ,\mathbf{\alpha_n})^T$,
here $\mathbf{\alpha_1},\dots ,\mathbf{\alpha_m}$ are the
least generating elements of $\Theta$, and
$\mathbf{\alpha_{m+1}},\dots ,\mathbf{\alpha_n}\in \mathbb Z^n$
are fundamental solutions of the linear equations
$B\mathbf{x}=\mathbf{0}$, then $P_1B^T=\textbf{0}$.
Let
$$
R=\begin{pmatrix}
 2 & ~1 & \dots & ~1 \\
 1 & ~2 & \dots & ~1 \\
 \vdots & \vdots & ~ & \vdots \\
 1 & ~1 & \dots & ~2
 \end{pmatrix}, \quad
 P= \begin{pmatrix}
 P_2 \\
 P_1
 \end{pmatrix},
$$
where $P_2$ is a $m\times n$ sub-matrix of $R$ such that $P$ is
invertible. Then $P_1=(\textbf{0},E_{n-m})P$, i.e.,
$P_1P^{-1}=(\mathbf{0},E_{n-m})$, where $E_{n-m}$ denotes the $n-m$
order unit matrix.
Let
\begin{equation}\label{tau}
\mathbf{\tau}
=(\mathbf{\tau_1},\dots ,\mathbf{\tau_m})=P^{-1}
 \begin{pmatrix}
 E_m \\
 \mathbf{0}
 \end{pmatrix},
\end{equation}
then $\mathbf{\tau_1},\dots ,\mathbf{\tau_m}\in \mathbb Q^n$, and
they are linear independent. Furthermore,
$$
P_1\mathbf{\mathbf{\tau}}=P_1P^{-1}
 \begin{pmatrix}
 E_m \\
 \mathbf{0}
 \end{pmatrix}
= \begin{pmatrix}
 \mathbf{0} & E_{n-m}
 \end{pmatrix}
  \begin{pmatrix}
 E_m \\
 \mathbf{0}
 \end{pmatrix}
={\mathbf0},
$$
therefore $\mathbf{\tau_1},\dots ,\mathbf{\tau_m}$ are fundamental solutions
of the linear equations
\begin{equation}\label{p1x0}
P_1\textbf{x}=\textbf{0}.
\end{equation}

(1) For any $\mathbf{k}\in\Theta$, there exist $b_1,\dots ,b_m\in \mathbb R$,
such that $\mathbf{k}=b_1\mathbf {\alpha_1}+\dots +b_m\mathbf {\alpha_m}$.
Therefore
$$
P_1\mathbf{k}=b_1P_1\mathbf {\alpha_1}+\dots +b_mP_1\mathbf
{\alpha_m}=P_1B^T(b_1,\dots ,b_m)^T=\mathbf{0}.
$$
This means that $\mathbf{k}$ is a solution of \eqref{p1x0}, thus
there exists $\mathbf a=(a_1,\dots ,a_m)^T\in \mathbb Q^m$, such
that
$$
\mathbf{k}=a_1\mathbf{\tau_1}+\dots +a_m\mathbf{\tau_m}.
$$

(2) For any $\mathbf{k}\in \Theta'$, by (1), there exists
$\mathbf{a}=(a_1,\dots ,a_m)^T\in \mathbb Q^m$, such that
$\mathbf{k}=a_1\mathbf{\tau_1}+\dots +a_m\mathbf{\tau_m}$.
By \eqref{tau}, we have
$$
\mathbf{k}=\tau \mathbf a
 =P^{-1} \begin{pmatrix}
 E_m \\
 \mathbf{0}
 \end{pmatrix}
\mathbf a,
$$
thus
$$
 \begin{pmatrix}
 \mathbf a \\
 \mathbf{0}
 \end{pmatrix}
=P\mathbf{k}= \begin{pmatrix}
 P_2 \\
 P_1
 \end{pmatrix}
\mathbf{k}= \begin{pmatrix}
 P_2\mathbf{k} \\
 \mathbf{0}
 \end{pmatrix}.
$$
According to the choice of $P_2$, we obtain that $(a_1,\dots ,
a_m)^T\in\mathbb Z^m_+$, and $|\mathbf a|\geq 1$.
\end{proof}

\begin{lemma}\label{l2}
Assume {\rm (H1)} holds, and $\mathbf{\tau_1},\dots ,\mathbf{\tau_m}$
be given in Lemma \ref{lem1}. Then under the change of variables
 \begin{equation}\label{2.8}
\begin{gathered}
 z_i=y_1^{\tau_{i1}}\dots y_n^{\tau_{in}}, \quad i=1,\dots , m, \\
 w_j=y_{m+j}, \quad j=1,\dots , n-m,
 \end{gathered}
\end{equation}
system \eqref{2.7} becomes
\begin{equation}\label{2.9}
\begin{gathered}
 \dot z_i=z_i\bar F^i(\mathbf z), \quad i=1,\dots ,m,\\
 \dot w_{j}=\lambda_{m+j}w_j+w_j\bar F^{m+j}(\mathbf z), \quad j=1,\dots ,n-m,
 \end{gathered}
\end{equation}
where $\mathbf z=(z_1,\dots ,z_m)$, $\bar F^i(\mathbf z)~(i=1,\dots ,n)$
are formal power series with respect to $\mathbf z$, $\bar F^i(\mathbf{0})=0$.
\end{lemma}

\begin{proof}
By Lemma \ref{lem1} and \eqref{tildeF}, we have
\begin{equation}\label{fei1}
 \begin{gathered}
\dot z_i =z_i\Big(\tau_{i1}\frac{\tilde F^1(\mathbf{y})}{y_1}+\dots
 +\tau_{in}\frac{\tilde F^n(\mathbf{y})}{y_n}\Big),\quad i=1,\dots , m,\\
\dot w_j=\lambda_{m+j}w_j+w_j\frac{\tilde F^{m+j}(\mathbf{y})}{y_{m+j}},
\quad j=1,\dots , n-m,
 \end{gathered}
\end{equation}
where
\begin{equation}\label{fzhan}
\frac{\tilde F^i(\mathbf{y})}{y_i}
=\sum_{r=2}^{\infty}\sum_{|\mathbf{k}|=r,\langle\mathbf{k},\Lambda\rangle
=\lambda_i}\tilde F^i_{\mathbf{k}}y_1^{k_1}\dots y_i^{k_i-1} \dots y_n^{k_n},
\quad i=1,\dots ,n.
\end{equation}
Obviously, for every monomial $y_1^{k_1} \dots y_i^{k_i-1} \dots
y_n^{k_n}$ in the above expression, the exponents $ (k_1,\dots , k_i-1, \dots , k_n)
\in \Theta'$. By Lemma \ref{lem1}, there exist
$\mathbf{a}=(a_1,\dots ,a_m)^T$ in $\mathbb Z^m_+$ with $|\mathbf{a}|\geq 1$, such that
$$
(k_1,\dots , k_i-1,\dots , k_n)^T=a_1\mathbf{\tau_{1}}+\dots +a_m\mathbf{\tau_m}.
$$
Therefore
\begin{equation}
\begin{aligned}
\frac{\tilde F^i(\mathbf{y})}{y_i}
&=\sum_{r=2}^{\infty}\sum_{|\mathbf{k}|=r,\langle\mathbf{k},\Lambda\rangle=\lambda_i}
\tilde F^i_{\mathbf{k}}y_1^{a_1\tau_{11}+\dots +a_m\tau_{m1}}\dots y_n^{a_1\tau_{1n}
+\dots +a_m\tau_{mn}} \\
&=\sum_{r=2}^{\infty}\sum_{|\mathbf{k}|=r,\langle\mathbf{k},\Lambda\rangle
 =\lambda_i}\tilde F^i_{\mathbf{k}}(y_1^{\tau_{11}}\dots
y_n^{\tau_{1n}})^{a_1}\dots (y_1^{\tau_{m1}}\dots y_n^{\tau_{mn}})^{a_m} \\
&=\sum_{r=1}^{\infty}\sum_{|\mathbf{a}|=r} \bar F^i_{\mathbf a}z_1^{a_1}
 \dots z_m^{a_m},\quad i=1,\dots ,n.
\end{aligned} \label{fei3}
\end{equation}
By \eqref{fei1} and \eqref{fei3}, we can get \eqref{2.9}, and the lemma
is proved.
\end{proof}

\begin{remark} \label{rmk2.6} \rm
It should be pointed out that there are similar arguments in \cite{Bruno65, Bruno89}
 as used in Lemma \ref{lem1} and Lemma \ref{l2}. Here, to ensure that
$\bar F^i(\mathbf z)~(i=1,\dots ,n)$ in \eqref{2.9} are formal power series
with respect to $\mathbf z$, we give a different way to calculate
$\mathbf{\tau_1}, \dots , \mathbf{\tau_m}$.
\end{remark}



\begin{lemma}\label{l4}
Assume that {\rm (H1)} holds. If system \eqref{xaf} has a nontrivial
first integral, then system \eqref{2.9} has a nontrivial formal
first integral which is independent with $w_1, \dots , w_{n-m}$, and the system
\begin{equation}\label{2.10}
\dot{ \mathbf z}=\mathbf z\bar F(\mathbf z)
\end{equation}
has a nontrivial formal first integral, where
$\mathbf z\bar F(\mathbf z):= (z_1\bar F^1(\mathbf z), \dots , z_m\bar
F^m(\mathbf z))$.
\end{lemma}

\begin{proof}
Assume that \eqref{xaf} has a nontrivial formal first
integral $\Phi(\mathbf{x})$, then by Lemma \ref{l3},
 $\tilde \Phi(\mathbf{y})=\Phi(\mathbf{y}+h(\mathbf{y}))$ is a formal first
integral of system \eqref{2.7}, and $\tilde \Phi (\mathbf{y})$ is resonant
 with respect to $\Lambda$, therefore $\tilde \Phi(\mathbf{y})$ can be
written as
\begin{equation}\label{phizhan}
\tilde\Phi(\mathbf{y})=\sum_{i=1}^{\infty}\sum_{|\mathbf{k}|
=i,\langle\mathbf{k},\Lambda\rangle=0} \tilde\Phi_{\mathbf{k}}y_1^{k_1}\dots y_n^{k_n},
\end{equation}
where $\tilde \Phi_{\mathbf{k}}$ are nonzero constants.

Note that for every monomial $y_1^{k_1}\dots y_n^{k_n}$ in
\eqref{phizhan}, $(k_1,\dots , k_n)^T \in\Theta'$. By
Lemma \ref{lem1}, there exists $\mathbf{a}=(a_1,\dots ,a_m)^T\in \mathbb Z_+^m$,
$|\mathbf{a}|\geq 1$, such that
$$
(k_1,\dots , k_n)^T=a_1\mathbf{\tau_{1}}+\dots + a_m\mathbf{\tau_m}.
$$
Thus
\begin{align*}
\tilde\Phi(\mathbf{y})
&= \sum_{i=1}^{\infty}\sum_{|\mathbf{k}|=i,\,
\langle\mathbf{k},\Lambda\rangle=0} \tilde\Phi_{\mathbf{k}}y_1^{a_1\tau_{11}
 +\dots +a_m\tau_{m1}}\dots y_n^{a_1\tau_{1n}+\dots +a_m\tau_{mn}}\\
&= \sum_{i=1}^{\infty}\sum_{|\mathbf{k}|=i,\,
\langle\mathbf{k},\Lambda\rangle=0}\tilde\Phi_{\mathbf{k}}(y_1^{\tau_{11}}\dots
y_n^{\tau_{1n}})^{a_1}\dots (y_1^{\tau_{m1}}\dots y_n^{\tau_{mn}})^{a_m}\\
&= \sum_{i=1}^{\infty}\sum_{|\mathbf{k}|=i,\, \langle\mathbf{k},\Lambda\rangle=0}
 \tilde\Phi_{\mathbf{k}}z_1^{a_1}\dots z_m^{a_m}
=: \Psi(\mathbf z).
\end{align*}

Since \eqref{2.7} is changed to \eqref{2.9} under the transformation
\eqref{2.8}, $\Psi(\mathbf z)$ is a first
integral of \eqref{2.9}. It is clearly that $\Psi(\mathbf z)$ is
independent with $w_1,\dots ,w_{n-m}$, so it is also a
first integral of \eqref{2.10}.
\end{proof}

Expand $\bar F(\mathbf z)$ as
\begin{equation*}\label{2.11}
\bar F(\mathbf z)=\bar F_p(\mathbf z)+\bar F_{p+1}(\mathbf z)+\dots ,
\end{equation*}
where $p\geq 1, \bar F_k(\mathbf z)(k=p, p+1, \dots )$ are the $k$-th order
homogeneous polynomial. We can get the following lemma easily.

\begin{lemma}\label{lem26}
If system \eqref{2.10} has a nontrivial formal first integral $\Psi(\mathbf z)$,
then the lowest order homogeneous terms $\Psi_{q}(\mathbf z)$ is a first
integral of system
\begin{equation}\label{2qi}
\dot {\mathbf z}=\mathbf z\bar F_p(\mathbf z).
\end{equation}
\end{lemma}

System \eqref{2qi} can be treated as a quasi-homogeneous system of
degree $p+1$ with the exponents $s_1=\dots =s_m=1$ (for more
detail, see \cite{Furta}). Let $\mathbf {\xi}$ be a balance
of vector field $\mathbf z\bar F_p(\mathbf z)$, i.e., $\mathbf \xi$
is a nonzero solution of the algebraic equations
$\frac{1}{p}\mathbf \xi+\mathbf \xi\bar F_p(\mathbf \xi)=0$,
then system \eqref{2qi} has a particular solution
$\mathbf z_0(t)=t^{-\frac{1}{p}E_m}\mathbf \xi$.

We make the change of variables
$$
\mathbf z(t)=t^{-\frac{1}{p}E_m}(\mathbf \xi+\mathbf u),\quad s=\ln t,
$$
then system \eqref{2qi} reads
$$
\frac{d\mathbf u}{ds}=K\mathbf u +\hat{F}(\mathbf u),
$$
where
$$
K= \begin{pmatrix}
 \frac{1}{p}+\bar F_p^1(\mathbf \xi) &  &  \\
  & \ddots &  \\
  &  & \frac{1}{p}+\bar F_p^m(\mathbf \xi)
 \end{pmatrix}
+ \begin{pmatrix}
 \xi_1\frac{\partial \bar F_p^1(\mathbf \xi)}{\partial z_1} & \dots
& \xi_1\frac{\partial \bar F_p^1(\mathbf \xi)}{\partial z_m} \\
 \vdots & \dots & \vdots \\
 \xi_m\frac{\partial \bar F_p^m(\mathbf \xi)}{\partial z_1}
& \dots & \xi_m\frac{\partial \bar F_p^m(\mathbf \xi)}{\partial z_m} \\
 \end{pmatrix}
$$
is the so-called Kovalevskaya matrix associated to the balance
$\mathbf \xi$,
$$
\hat F(\mathbf u)=\frac{1}{p}E_m\mathbf \xi+(\mathbf \xi +\mathbf u)
\bar F_p(\mathbf \xi+\mathbf u)-\frac{\partial \mathbf z\bar
F_p(\mathbf z)}{\partial \mathbf z}\mathbf u
\Big |_{\mathbf z=\mathbf \xi}=O(\|\mathbf u\|^2).
$$
Now we can state our main result.

\begin{theorem}\label{zhuth}
Assume that {\rm (H1)} holds, $\mathbf \xi$ is a balance of vector field
$\mathbf z\bar F_p(\mathbf z)$, $\mathbf \mu=(\mu_1,\dots ,\mu_m)$ is
the eigenvalues of Kovalevskaya matrix
associated to the balance $\mathbf \xi$. Let
$$
\Omega=\big\{\mathbf{k}=(k_1,\dots ,k_m)^T\in\mathbb Z_{+}^m:
\langle \mathbf{k},\mathbf \mu\rangle=0\big\}.
$$
\begin{itemize}
\item[(1)] If rank $\Omega=0$, then system \eqref{xaf} does not have any
 nontrivial formal first integrals in the neighborhood of $\mathbf{x}=\mathbf{0}$;

\item[(2)] If rank $\Omega=l>0$, then system \eqref{xaf} has at most
$l$ functionally independent formal first integrals in the
neighborhood of $\mathbf{x}=\mathbf{0}$.
\end{itemize}
\end{theorem}

\begin{proof}
(1) Assume that system \eqref{xaf} admits a formal first integral in a
neighborhood of $\mathbf{x}=\mathbf{0}$, then by Lemma \ref{l4},
system \eqref{2.10} admits a formal first integral, and by
Lemma \ref{lem26}, system \eqref{2qi} has a homogeneous
first integral. According to \cite[Theorem 1]{Furta}, we know that
$\mu_1,\dots ,\mu_m$ are $\mathbb N$-resonant, which contradicts
with rank $\Omega=0$.

(2) Assume that system \eqref{xaf} admits $l+1$ functional independent
first integrals $\Phi^1(\mathbf{x})$, $\dots $,
$\Phi^{l+1}(\mathbf{x})$ in a neighborhood of $\mathbf{x}=\mathbf{0}$.
By Lemma \ref{l3}, Lemma \ref{l2} and
 Lemma \ref{l4}, after a sequence of transformations, we can see that system
\eqref{2.10} have $l+1$ formal first integrals
$\Psi^1(\mathbf z),\dots ,\Psi^{l+1}(\mathbf z)$.
Since the sequence of transformations which transform \eqref{xaf} to
\eqref{2.10} are local invertible, hence
$\Psi^1(\mathbf z),\dots ,\Psi^{l+1}(\mathbf z)$ are functional independent in the
neighborhood of $\mathbf z=\mathbf{0}$. According to Ziglin lemma in \cite{Ziglin},
system \eqref{2.10} has $l+1$ first integrals
$\tilde \Psi^1(\mathbf z),\dots ,\tilde\Psi^{l+1}(\mathbf z)$ whose lowest order
homogeneous terms $\tilde\Psi^1_{q_1}(\mathbf z),\dots ,
\tilde\Psi_{q_{l+1}}^{l+1}(\mathbf z)$ are functionally independent, and by
Lemma \ref{lem26}, $\tilde\Psi^1_{q_1}(\mathbf z),\dots ,\tilde\Psi_{q_{l+1}}^{l+1}
(\mathbf z)$ are $l+1$ first integrals of
 quasi-homogeneous system \eqref{2qi}. By \cite[Theorem B]{kwek}, we get
rank $\Omega\geq l+1$, which
 contradicts with the assumption of theorem. The proof is complete.
\end{proof}

\section{Example}

Based on the arguments in Section 2, one can give an algorithm
to test whether a given system admits formal first integral or not, see
the next example.

\begin{example} \rm
Consider the  generalized Lotka-Volterra system
\begin{equation}\label{lizi1}
 \begin{gathered}
 \dot x_1=x_1+a_1x_1x_1x_2+a_2x_1x_3x_4+a_3x_1x_5x_6, \\
 \dot x_2=-x_2+b_1x_2x_1x_2+b_2x_2x_3x_4+b_3x_2x_5x_6, \\
 \dot x_3=\sqrt 2x_3+c_1x_3x_1x_2+c_2x_3x_3x_4+c_3x_3x_5x_6, \\
 \dot x_4=-\sqrt 2x_4+d_1x_4x_1x_2+d_2x_4x_3x_4+d_3x_4x_5x_6, \\
 \dot x_5=\sqrt 3x_5+e_1x_5x_1x_2+e_2x_5x_3x_4+e_3x_5x_5x_6, \\
 \dot x_6=-\sqrt 3x_6+f_1x_6x_1x_2+f_2x_6x_3x_4+f_3x_6x_5x_6,
 \end{gathered}
\end{equation}
where $a_i$, $b_i$, $c_i$, $d_i$, $e_i$, $f_i\in\mathbb{R}$,
$i=1,2,3$.
Lotka-Volterra equations can be used to describe cooperation and competition
between biological species in ecology. In 1988,
Brenig \cite{Brenig1} introduced generalized Lotka-Volterra
equations. By transforming the original equations into a canonical form,
he discussed the integrability of
some type generalized Lotka-Volterra equations. As an application, we consider
the nonexistence of formal first integral
for above Lotka-Volterra system.
\end{example}

Obviously, the eigenvalues of Jocabi matrix at $\mathbf{x}=\mathbf{0}$
of system \eqref{lizi1} are
$$
\lambda_1=1, \quad \lambda_2=-1, \quad \lambda_3=\sqrt 2,\quad
\lambda_4=-\sqrt 2,\quad  \lambda_5=\sqrt 3,\quad  \lambda_6=-\sqrt 3,
$$
and system \eqref{lizi1} is a normal form. Let
\begin{gather*}
 \Omega=\big\{\mathbf{k}=(k_1,\dots ,k_{6})^T\in\mathbb Z^{6}_+:
\sum_{i=1}^{6}k_i\lambda_i=0\big\}, \\
 \Theta=\big\{\mathbf{k}=(k_1,\dots ,k_{6})^T\in\mathbb Z^{6}:
\sum_{i=1}^{6}k_i\lambda_i=0\big\},
\end{gather*}
then $\operatorname{rank} \Omega=\operatorname{rank} \Theta=3$, and
$(1,1,0,0,0,0)^T$, $(0,0,1,1,0,0)^T$, $(0,0,0,0,1,1)^T$ are the least
generating elements of $\Theta$. Making the change of variables
$$
z_1=x_1x_2,\quad z_2=x_3x_4,\quad  z_3=x_5x_6,\quad w_1=x_2,\quad
 w_2=x_4,\quad w_3=x_6,
$$
 system \eqref{lizi1} becomes
\begin{equation} \label{lizijia}
 \begin{gathered}
 \dot z_1=z_1(\alpha_{1}z_1+\alpha_{2}z_2+\alpha_{3}z_3), \\
 \dot z_2=z_2(\beta_{1}z_1+\beta_{2}z_2+\beta_{3}z_3), \\
 \dot z_3=z_3(\gamma_{1}z_1+\gamma_{2}z_2+\gamma_{3}z_3), \\
 \dot w_1=-w_1+b_1w_1z_1+b_2w_1z_2+b_3w_1z_3, \\
 \dot w_2=-\sqrt 2w_2+e_1w_2z_1+e_2w_2z_2+e_3w_2z_3, \\
 \dot w_3=-\sqrt 3w_3+f_1w_3z_1+f_2w_3z_2+f_3w_3z_3, 
 \end{gathered}
\end{equation}
where $\alpha_{i}=a_i+b_i$, $\beta_{i}=c_i+d_i$,
$\gamma_{i}=e_i+f_i$, $i=1,2,3$.

By Lemma \ref{lem26}, we need only to investigate the
formal first integral for the system
\begin{equation}\label{lizi2}
\begin{gathered}
 \dot z_1=z_1(\alpha_1z_1+\alpha_2z_2+\alpha_3z_3), \\
 \dot z_2=z_2(\beta_1z_1+\beta_2z_2+\beta_3z_3), \\
 \dot z_3=z_3(\gamma_1z_1+\gamma_2z_2+\gamma_3z_3). \\
 \end{gathered}
\end{equation}
System \eqref{lizi2} is a quasi-homogeneous system of degree $2$ with 
exponents $s_1=s_2=s_3=1$. Assume
$\alpha_1\neq 0$, then $\xi=(-\frac{1}{\alpha_1},0,0)^T$ is a balance of
system \eqref{lizi2}.
Making the change of variables
\begin{equation}\label{lizi2-1}
 z_1=t^{-1}(-\frac{1}{\alpha_1}+u_1), \quad
 z_2=t^{-1}u_2, \quad
 z_3=t^{-1}u_3, \quad
 t=e^{s}, 
\end{equation}
system \eqref{lizi2} reads
\begin{equation}\label{lizi3}
 \begin{gathered}
 u'_1=-u_1-\frac{\alpha_2}{\alpha_1}u_2-\frac{\alpha_3}{\alpha_1}u_3
 +u_1(\alpha_1u_1+\alpha_2u_2+\alpha_3u_3), \\
 u'_2=(1-\frac{\beta_1}{\alpha_1})u_2+u_2(\beta_1u_1+\beta_2u_2+\beta_3u_3), \\
 u'_3=(1-\frac{\gamma_1}{\alpha_1})u_3+u_3(\gamma_1u_1+\gamma_2u_2+\gamma_3u_3), 
 \end{gathered}
\end{equation}
where $'$ means the derivative with respect to $s$.
Here the corresponding Kovalevskaya matrix is
$$
K= \begin{pmatrix}
 -1 & -\frac{\alpha_2}{\alpha_1} & -\frac{\alpha_3}{\alpha_1} \\
 0 & 1-\frac{\beta_1}{\alpha_1} & 0 \\
 0 & 0 & 1-\frac{\gamma_1}{\alpha_1} \\
 \end{pmatrix}.
$$
Obviously,
$$
\mu_1=-1,\quad \mu_2=1-\frac{\beta_1}{\alpha_1},\quad
\mu_3=1-\frac{\gamma_1}{\alpha_1}
$$
are eigenvalues of $K$.
Let
$$
\Omega_1=\big\{\mathbf{k}=(k_1,k_2,k_3)^T\in\mathbb
Z_+^3:\sum_{i=1}^3k_i\mu_i=0\big\}.
$$
By Theorem \ref{zhuth}, we have the following results.

\begin{theorem}\label{th31}
Assume $\alpha_1\neq 0$.
\begin{itemize}
\item[(1)] If rank $\Omega_1=0$, then system \eqref{lizi1} does not have
any nontrivial formal first integrals in the neighborhood of
$\mathbf{x}=\mathbf{0}$;

\item[(2)] If rank $\Omega_1=l_1>0$, then system \eqref{lizi1} has at most
$l_1$ functionally independent formal first integrals in the
neighborhood of $\mathbf{x}=\mathbf{0}$.
\end{itemize}
\end{theorem}

It is not difficult to see that if $\alpha_{1},\beta_{1},\gamma_{1}$ are
$\mathbb Z$-non-resonant, and rank $\Omega_1=0$,
by Theorem \ref{th31}, we get nonexistence of formal first
integrals in the neighborhood of $\mathbf{x}=\mathbf{0}$.
Furthermore, we can obtain following conclusions.

\begin{corollary} \label{coro33}
\begin{itemize}
 \item[(1)] If $\alpha_1$, $\beta_1$, $\gamma_1$ are $\mathbb Z$-non-resonant,
then system \eqref{lizi1} does not have any nontrivial formal first
integrals in the neighborhood of $\mathbf{x}=\mathbf{0}$;

 \item[(2)] If $\alpha_1=\beta_1=\gamma_1\ne 0$, $\alpha_2=\alpha_3=0$
and $\frac{\gamma_3}{\beta_2}\not\in\mathbb{Q}$, then system
\eqref{lizi1} does not have any nontrivial formal first integrals in
the neighborhood of $\mathbf{x}=\mathbf{0}$.

 \item[(3)] If $\alpha_1=\beta_1=\gamma_1\ne 0$, $\alpha_2=\alpha_3=0$
and $\frac{\gamma_3}{\beta_2}\in\mathbb{Q}$, then system \eqref{lizi1}
has at most one nontrivial formal first integral in the neighborhood
of $\mathbf{x}=\mathbf{0}$.
\end{itemize}
\end{corollary}

\begin{proof}
The first case is obvious, we omit it.
Let us consider the cases when $\alpha_1=\beta_1=\gamma_1\ne 0$, and
$\alpha_2=\alpha_3=0$. In this situation, we need to consider
\begin{equation}\label{lizi33}
 \begin{gathered}
 u'_1=-u_1+\alpha_1u_1^2, \\
 u'_2=u_2(\alpha_1u_1+\beta_2u_2+\beta_3u_3), \\
 u'_3=u_3(\alpha_1u_1+\gamma_2u_2+\gamma_3u_3), \\
 u'_0=-u_0,
 \end{gathered}
\end{equation}
where $u_0=e^{-s}$. It is clearly that
$\mu_1=-1$, $\mu_2=0$, $\mu_3=0$,
and $\operatorname{rank}\Omega_1=2>0$.
Let
$$
\Theta_1=\big\{\mathbf{k}=(k_1,k_2,k_3,k_4)^T\in\mathbb
Z^4 : \sum_{i=1}^4k_i\mu_i=0\big\},
$$
where $\mu_4=-1$.

By the Poincar\'e-Dulac normal form theory, system \eqref{lizi33} can be
reduced to the  canonical form
\begin{equation}\label{lizi4}
 \begin{gathered}
 u'_1=-u_1+u_1\varphi^1(u_2,u_3), \\
 u'_2=u_2(\beta_2u_2+\gamma_3u_3)+\varphi^2(u_2,u_3), \\
 u'_3=u_3(\gamma_2u_2+\gamma_3u_3)+\varphi^3(u_2,u_3), \\
 u'_0=-u_0,
 \end{gathered}
\end{equation}
where $\varphi^1(u_2,u_3)=O(|(u_2, u_3)|)$, 
$\varphi^2(u_2,u_3)=O(|(u_2,u_3)|^2)$ and 
$\varphi^3(u_2,u_3)=O(|(u_2, u_3)|^3)$. The subsystem of
system \eqref{lizi4},
\begin{gather*}
 u'_2=u_2(\beta_2u_2+\beta_3u_3)+\varphi^2(u_2,u_3),\\
 u'_3=u_3(\gamma_2u_2+\gamma_3u_3)+\varphi^3(u_2,u_3) \\
\end{gather*}
is a semi-quasihomogeneous system with exponents $s_1=s_2=1$, and
its quasi-homogeneous cut is
\begin{equation}\label{lizi5}
 \begin{gathered}
 u'_2=u_2(\beta_2u_2+\beta_3u_3), \\
 u'_3=u_3(\gamma_2u_2+\gamma_3u_3). 
 \end{gathered}
\end{equation}
Note that if $\beta_2\neq 0$, then $(-\frac{1}{\beta_2},0)$ is a balance
of \eqref{lizi5}, and the corresponding Kovalevskaya matrix is
$$
K'= \begin{pmatrix}
 -1 & -\frac{\beta_3}{\beta_2} \\
 0 & 1-\frac{\gamma_3}{\beta_2} 
 \end{pmatrix}.
$$
Obviously, $\bar\mu_1=-1$,
$\bar\mu_2=1-\frac{\gamma_{3}}{\beta_{2}}$ are eigenvalues of
$K'$. Let
$$
\Omega_2=\big\{\mathbf{k}=(k_1,k_2)^T\in\mathbb
Z^2_+: k_1\bar\mu_1+k_2\bar\mu_2=0\big\}.
$$

By Theorem \ref{zhuth}, we know that if $\operatorname{rank}\Omega_2=0$,
then \eqref{lizi1} does not have any nontrivial formal first
integrals in the neighborhood of $\mathbf{x}=\mathbf{0}$; if rank
$\Omega_2=l_2>0$, then \eqref{lizi1} has at most $l_2$ functionally
independent formal first integrals in the neighborhood of 
$\mathbf{x}=\mathbf{0}$. Therefore we get the proofs of last two cases.
\end{proof}

\begin{remark} \label{rmk34} \rm
If $\Omega_2=l_2>0$, one can use the same idea as the change of variables
 \eqref{lizi2-1} to get a new system like \eqref{lizi3} to do more investigations.
 While, one particular case should be noted is that, if there exist 
$i_0\in\mathbb N$, such that for every $j>i_0$, 
$\operatorname{rank} \Theta_j= \operatorname{rank}\Theta_{i_0}=l_{i_0}\geq 2$, 
we can not get the nonexistence of formal first integral for system
\eqref{lizi1}. And this case always implies the partial existence of formal 
first integral for \eqref{lizi1}, i.e.,
\eqref{lizi1} may have $l_{i_0}-1$  formal first integrals in a neighborhood 
of $\mathbf{x}=\mathbf{0}$.
\end{remark}

\subsection*{Acknowledgments}
This work is supported by NSFC grants (No. 11771177, 11501242, 11301210),
by the China Automobile Industry Innovation and Development Joint Fund
 (No. U1664257), by Program for Changbaishan Scholars of Jilin Province
and Program for JLU Science, Technology Innovative Research Team
(No. 2017TD-20), SF (20170520055JH, 20140520053JH) of Jilin, China
 and by the ESF (JJKH20160398KJ,JJKH20170776KJ) of Jilin, China.


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\end{document}