Limit cycles appearing from the perturbation of a system with a multiple line of critical points

Consider the planar ordinary differential equation ẋ = −y(1 − y)m, ẏ = x(1 − y)m, where m is a positive integer number. We study the maximum number of zeroes of the Abelian integral M that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field. One of the key points of our approach is that we obtain a simple expression of M based on some successive reductions of the integrals appearing during the procedure. MSC 2010. Primary: 34C08. Secondary: 34C07, 34C23, 37C27.