  The dynamics of a beam held on a horizontal frame by springs and bouncing off
a step is described by a separable two degrees of freedom Hamiltonian system
with impacts that respect, point wise, the separability symmetry. The energy in
each degree of freedom is preserved, and the motion along each level set is
conjugated, via action angle coordinates, to a geodesic flow on a flat
two-dimensional surface in the four dimensional phase space. Yet, for a range
of energies, these surfaces are not the simple Liouville-Arnold tori - these
are tori of genus two, thus the motion on them is not conjugated to simple
rotations. Namely, even though energy is not transferred between the two
degrees of freedom, the impact system is quasi-integrable and is not of the
Liouville-Arnold type. In fact, for each level set in this range, the motion is
conjugated to the well studied and highly non-trivial dynamics of directional
motion in L-shaped billiards, where the billiard area and shape as well as the
direction of motion vary continuously on iso-energetic level sets. Return maps
to Poincar\'e section of the flow are shown to be conjugated, on each level
set, to interval exchange maps which are computed, up to quadratures, in the
general nonlinear case and explicitly for the case of two linear oscillators
bouncing off a step. It is established that for any such oscillator-step system
there exist step locations for which some of the level sets exhibit motion
which is neither periodic nor ergodic. Changing the impact surface by
introducing additional steps, staircases, strips and blocks from which the
particle is reflected, leads to iso-energy surfaces that are foliated by
families of genus-k level set surfaces, where the number and order of families
of genus k depend on the energy.
