Topological analysis of a billiard in elliptic ring in a potential field

We study a billiard in a domain bounded by two confocal ellipses. The Hooke potential is placed at the center of the ellipses. This dynamic system turns out to be Liouville integrable. Therefore, we can make a topological analysis studying the foliation of the phase manifold by integrals. We calculate Fomenko–Zieschang invariants (marked molecules) for isoenergy manifolds of every level of the Hamiltonian, and also give examples of other integrable systems that are Liouville equivalent to our billiard system.