Integrability Conditions for Complex Homogeneous Kukles Systems

Abstract

In this paper we study the existence of local analytic first integrals for complex polynomial differential systems of the formẋ = x + P n (x, y),ẏ = −y, where P n (x, y) is a homogeneous polynomial of degree n, called the complex homogeneous Kukles systems of degree n. We characterize all the homogeneous Kukles systems of degree n that belong to the Sibirsky ideal. Finally, we provide necessary and sufficient conditions when n = 2, . . . , 7 in order that the complex homogeneous Kukles system has a local analytic first integral computing the saddle constants and using Gröbner bases to find the decomposition of the algebraic variety into its irreducible components.