  In this text we study billiards on ovals and investigate some consequences of
a rotational symmetry of the boundary on the dynamics. As it simplifies some
calculations, the symmetry helps to obtain the results. We focus on periodic
orbits with the same symmetry of the boundary which always exist and prove that
in general half of them are elliptic and Moser stable and the other half are
hyperbolic with homo(hetero)clinic intersections.
