The displacement map associated to polynomial perturbations of some nongeneric Hamiltonians

It is known that the Principal Poincar'e Pontryagin Function is generically an Abelian integral. In non generic cases it is an iterated integral. In previous papers one of the authors gives a precise description of the Principal Poincar'e Pontryagin Function, an iterated integral af length at most 2, involving a logarithmic function with only one ramification at a point at infinity. We show here that this property can be generalized to Hamiltonians having real points at infinity and satisfying some properties.