Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

Abstract

This paper study the type of integrability of differential systems with separable variables $\dot{x}=h\left (x\right )f\left (y\right )$, $\dot{y}= g\left (y\right )$, where $h$, $f$ and $g$ are polynomials. We provide a criterion for the existence of generalized analytic first integrals of such differential systems. Moreover we characterize the polynomial integrability of all such systems.

In the particular case $h\left (x\right )=\left (ax+b\right )^{m}$ we provide necessary and sufficient conditions in order that this subclass of systems has a generalized analytic first integral. These results extend known results from Giné et al. (Discrete Contin. Dyn. Syst. 33:4531–4547, 2013) and Llibre and Valls (Discrete Contin. Dyn. Syst., Ser. B 20:2657–2661, 2015). Such differential systems of separable variables are important due to the fact that after a blow-up change of variables any planar quasi-homogeneous polynomial differential system can be transformed into a special differential system of separable variables $\dot{x}=xf\left (y\right )$, $\dot{y}=g\left (y\right )$, with $f$ and $g$ polynomials.