Analytical integrability problem for perturbations of cubic Kolmogorov systems

We solve, by using normal forms, the analytical integrability problem for differential systems in the plane whose first homogeneous component is a cubic Kolmogorov system whose origin is 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), with 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.