In this paper, we Study small polynomial perturbations of a Hamiltonian vector field with Hamiltonian F formed by a product of (d + 1) real linear functions in two variables. We assume that the corresponding lines are in a general position in R(2). That is, the lines are distinct, non-parallel, no three of them have a common point and all critical values not corresponding to intersections of lines are distinct. We prove in this paper that the principal Poincare-Pontryagin function M(k)(t), associated to such a perturbation and to any family of ovals surrounding a singular point of center type, belongs to the C[t, 1/t]-module generated by Abelian integrals and some integrals I(i,j)*(t), with 1 <= i < j <= d defined in the paper. Moreover, I(i,j)*(t) are not Abelian integrals. They are iterated integrals of length two. (C) 2008 Elsevier Inc. All rights reserved.