Mel'nikov method revisited
  We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitable extension of a classical method, and which is parallel and -- at least in part -- complementary to the standard procedure. This approach can be also applied to some ``degenerate'' situations, as to the case of nonhyperbolic unstable points, or of critical points located at the infinity (thus giving rise to unbounded orbits, e.g. the Keplerian parabolic orbits), and it is naturally ``compatible'' with the presence of general symmetry properties of the problem. 
G. Cicogna and M. Santoprete