The Liouville foliation of nonconvex topological billiards

Together with the classical plane billiards, topological billiards can be considered, where the motion occurs on a locally flat surface obtained by isometrically gluing together several plane domains along their boundaries, which are arcs of confocal quadrics. A point moves inside each of the domains along straight line segments; when it reaches the boundary of a domain, it passes to another domain. Previously, the author gave a Liouville classification of all topological billiards obtained by gluing along convex boundaries. In the present paper, all topological integrable billiards obtained by gluing along convex or nonconvex boundaries from elementary billiards bounded by arcs of confocal quadrics are classified. For some of such nonconvex topological billiards, the Fomenko–Zieschang invariants (marked molecules $W^*$) for Liouville equivalence are calculated.