Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas

An integrable system that is a billiard in a domain bounded by confocal ellipses and hyperbolas is studied. This system arises in the description of the motion of a point inside this domain with natural reflection from the boundary. The topological invariant of Liouville equivalence of such systems, namely, the Fomenko–Zieschang molecule, is calculated using a new method developed by the author.