Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems

Abstract

In this paper we find necessary and sufficient conditions in order that a planar quasi-homogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasi-homogeneous polynomial differential system can be transformed into a differential system of the form u˙=uf(v), v˙=g(v) with f(v) and g(v) polynomials, and vice versa.

The first and second authors are partially supported by a MICINN/ FEDER grant number MTM2011-22877 and by a AGAUR (Generalitat de Catalunya) grant number 2009SGR 381. The third author is partially supported by a MICINN/ FEDER grant number MTM2008-03437, by a AGAUR grant number 2009SGR 410, by ICREA Academia and by FP7-PEOPLE-2012-IRSES- 316338 and 319888