  Near-integrability is usually associated with smooth small perturbations of
smooth integrable systems. Studying integrable mechanical Hamiltonian flows
with impacts that respect the symmetries of the integrable structure provides
an additional rich class of non-smooth systems that can be analyzed. Such
systems exhibit rich dynamics, as, in addition to the underlying integrable
structure, some of the trajectories may undergo only transverse impact, others
may undergo also tangential impacts, and some trajectories do not impact at
all. Under perturbations, each of these classes of orbits behaves differently.
Tools for classifying these different types of dynamics in 2 degrees-of-freedom
Hamiltonian impact systems with underlying separable integrable dynamics are
presented. Moreover, some of these tools may be extended to far from integrable
cases. In particular, a generalization of the energy momentum bifurcation
diagram, Fomenko graphs and the hierarchy of bifurcations framework to impact
systems is constructed. It is shown that such representations classify
dynamically different regions in phase space. For the integrable and near
integrable (small perturbations) cases these provide global information on the
dynamics whereas for the far from integrable (non-perturbative) regimes, these
provide rough classification of the first impact map. The interpretation of
these results in terms of projections of solutions to the configuration space
as well as the relations of these to the Hill region are presented.
