Symmetric periodic orbits in symmetric billiards

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.