We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equilibrium
of the saddle-center type can have a sequence of small parameter values for which a
one-round homoclinic orbit to this equilibrium exists. This contrasts with the well-known
findings by Amick and McLeod and others that solutions of such type do not exist in
analytic Hamiltonian systems, and that the separatrices are split by the exponentially
small quantity. We also discuss existence of homoclinic trajectories to small periodic
orbits of the Lyapunov family as well as symmetric periodic orbits near the homoclinic
connection. Our further result, illustrated by simulations, concerns the complicated
structure of orbits related to passage through a non-smooth bifurcation of a periodic
orbit.