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

Abstract

This paper study the type of integrability of the differential systems with separable variables ẋ = h (x) f (y), ẏ = g (y), 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 (x) = (ax + b) 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 [5] and [13]. 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 ẋ = xf (y), ẏ = g (y), with f and g polynomials.