Kowalevskaya exponents and normal forms of planar dynamical systems. Application to integrability problem

Abstract

In this work dynamical properties of the solutions of differential systems in the plane are provided using certain constants related with the Kowalevskaya exponents. An orbital normal form that allows to study the existence of analytic and non-analytic first integrals is given. We apply the results to study the algebraic and analytic integrability around several singularities.