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