Integrability of two dimensional quasi-homogeneous polynomial differential systems

Abstract

In terms of the conservative-dissipative de composition of a vector field, we characterize the two dimensional quasi-homogeneous polynomial differential systems with a polynomial first integral (in these systems, polynomial integrability and analytic integrability are equivalent). We also provide an easy method to allow us to compute them and their centers. Finally, as an application, we find the quasi- homogeneous polynomial systems of degree two which have a polynomial first integral (the cubic homogeneous systems, among others, are included).