Liouville Foliations of Topological Billiards with Slipping

In the paper, a new class of integrable billiards, namely, billiards with slipping, is studied. At the reflection from the boundary, a billiard particle of such a system may not only change its velocity, but also move some distance along the border. Some laws of slipping preserve the integrability of flat confocal and circular billiards and billiard books glued from them, i.e., billiards on cell complexes. In the paper, the topology of the Liouville foliations for several integrable billiards with slipping, both flat and locally flat, is studied. Two such systems are Liouville equivalent to integrable geodesics flows of small degrees on nonorientable two-dimensional surfaces, namely, the projective plane and the Klein bottle. This shows that the nonorientability of a two-dimensional surface, in itself, is not an obstacle to its implementability by an appropriate integrable billiard.