We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium
with a homoclinic loop. We identify the set of orbits which are homoclinic to
the center manifold of the equilibrium via a Lyapunov- Schmidt reduction
procedure. This leads to the study of a singularity which inherits certain
structure from the Hamiltonian nature of the system. Under non-degeneracy
assumptions, we classify the possible Morse indices of this singularity,
permitting a local description of the set of homoclinic orbits. We also
consider the case of time-reversible Hamiltonian systems.