  The ergodic properties of two uncoupled oscillators, a horizontal and
vertical one, residing in a class of non rectangular star-shaped polygons with
only vertical and horizontal boundaries and impacting elastically from its
boundaries are studied. We prove that the iso-energy level sets topology
changes non-trivially; the flow on level sets is always conjugated to a
translation flow on a translation surface, yet, for some segments of partial
energies the genus of the surface is strictly larger than one. When at least
one of the oscillators is un-harmonic, or when both are harmonic and
non-resonant, we prove that for almost all partial energies, including the
impacting ones, the flow on level sets is unique ergodic. When both oscillators
are harmonic and resonant, we prove that there exist intervals of partial
energies on which periodic ribbons and additional ergodic components co-exist.
We prove that for almost all partial energies in such segments the motion is
unique ergodic on the part of the level set that is not occupied by the
periodic ribbons. This implies that ergodic averages project to piecewise
smooth weighted averages in the configuration space.
