Analytic integrability of a class of planar polynomial differential systems

Abstract

In this paper we find necessary and sufficient conditions in order that the differential systems of the form $\dot x = x f(y)$, $\dot y =g(y)$, with $f$ and $g$ polynomials, have a first integral which is analytic in the variable $x$ and meromorphic in the variable $y$. We also characterize their analytic first integrals in both variables $x$ and $y$. These polynomial differential systems are important because after a convenient change of variables they contain all quasi--homogeneous polynomial differential systems in $\mathbb{R}^2$.