© 2018 Elsevier Ltd 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.