  A class of Hamiltonian impact systems exhibiting smooth near integrable
behavior is presented. The underlying unperturbed model investigated is an
integrable, separable, 2 degrees of freedom mechanical impact system with
effectively bounded energy level sets and a single straight wall which
preserves the separable structure. Singularities in the system appear either as
trajectories with tangent impacts or as singularities in the underlying
Hamiltonian structure (e.g. separatrices). It is shown that away from these
singularities, a small perturbation from the integrable structure results in
smooth near integrable behavior. Such a perturbation may occur from a small
deformation or tilt of the wall which breaks the separability upon impact, the
addition of a small regular perturbation to the system, or the combination of
both. In some simple cases explicit formulae to the leading order term in the
near integrable return map are derived. Near integrability is also shown to
persist when the hard billiard boundary is replaced by a singular, smooth,
steep potential, thus extending the near-integrability results beyond the scope
of regular perturbations. These systems constitute an additional class of
examples of near integrable impact systems, beyond the traditional one
dimensional oscillating billiards, nearly elliptic billiards, and the
near-integrable behavior near the boundary of convex smooth billiards with or
without magnetic field.
