  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.
