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It is known that the Principal Poincar\'e Pontryagin Function is genericallyan Abelian integral. We give a sufficient condition on monodromy to ensure thatit is an Abelian integral also in non generic cases. In non generic cases it isan iterated integral. Uribe [17, 18] gives in a special case a precisedescription of the Principal Poincar\'e Pontryagin Function, an iteratedintegral of length at most 2, involving logarithmic functions with only oneramification at a point at infinity. We extend this result to some nonisodromic families of real Morse polynomials.

Principal Poincar\'e Pontryagin Function associated to some families of Morse real polynomials