Analytical integrability problem for perturbations of cubic Kolmogorov systems

Abstract

We solve, by using normal forms, the analytic integrability problem for differential systems in the plane whose first homogeneous component is a cubic Kolmogorov system being the origin an isolated singularity. As an application, we give the analytically integrable systems of a class of systems x' = x(P2 + P3); y' = y(Q2 +Q3); being Pi;Qi homogeneous polynomials of degree i. We also prove that for any n>=3, there are analytically integrable perturbations of x' = xPn; y' = yQn which are not orbital equivalent to its first homogeneous component.