Integrability of Lotka-Volterra type systems of degree 4 (Q663661)

A system of the form \[ \begin{aligned} \dot x &=x+ f(x,y)= x+o(|x,y|),\\ \dot y &=-\lambda y+ g(x,y)=-\lambda y+ o(|x,y|)\end{aligned} \] is called to be integrable at the origin if and only if there exists an analytic change of coordinates, \[ (u,v)= (x+ o(|x,y|),\,y+ o(|x,y|)) \] which transforms the system to \[ \begin{aligned} \dot u &=uH(u, v)= u(1+ o(|x,y|)),\\ \dot v &=-\lambda vH(u,v)=-\lambda v(1+o(|x, y|)),\end{aligned} \] where \(H(u,v)\) is an analytic function in a neighborhood of the origin. The integrability of a Lotka-Volterra type system of the form \[ \begin{aligned} \dot x &=x(1+ a_1 x^3+ a_2 x^2 y+ a_3 xy^2+ a_4 y^3),\\ \dot y &=y(-\lambda+ b_1 x^3+ b_2 x^2 y+ b_3 xy^2+ b_4 y^3),\end{aligned} \] where \(\lambda\) is an integer \(\geq 2\), is studied. Some sufficient conditions for the integrability of such systems are proved and some necessary conditions are given by studying the first two saddle values of the system. Necessary and sufficient conditions for the integrability of systems with \(1:-2\), \(1:-5\), \(1:-8\) resonances are derived.