Artículo de publicación ISIIn this paper, we consider a small polynomial perturbation of the Hamiltonian vector field with the Hamiltonian F(x, y) = x[y2 − (x − 3)2] having a center bounded by a triangle. The main result of this work is that the principal Poincar´e–Pontryagin function associated with such a perturbation and with the family of ovals surrounding the center belongs to the C[t, 1/t] module generated by Abelian integrals I0(t) and I2(t) and by I∗(t), where I∗(t) is not an Abelian integral. We show that, in general, the principal Poincar´e– Pontryagin function of order two of a polynomial perturbation of the degree at least five is not an Abelian integral