  Motivated by black holes surrounded by accretion structures, we consider in
this series static and axially symmetric black holes "perturbed"
gravitationally as being encircled by a thin disc or a ring. In previous
papers, we employed several different methods to detect, classify and evaluate
chaos which can occur, due to the presence of the additional source, in
time-like geodesic motion. Here we apply the Melnikov-integral method which is
able to recognize how stable and unstable manifolds behave along the perturbed
homoclinic orbit. Since the method standardly works for systems with one degree
of freedom, we first suggest its modification applicable to two degrees of
freedom (which is the our case), starting from a suitable canonical
transformation of the corresponding Hamiltonian. The Melnikov function reveals
that, after the perturbation, the asymptotic manifolds tend to split and
intersect, consistently with the chaos found by other methods in the previous
papers.
