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

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