Isoenergy manifolds of integrable billiard books

We consider a class of integrable Hamiltonian systems with two degrees of freedom — billiard books, which are generalizations of billiards bounded by arcs of confocal quadrics. The first issue arising in the study of billiards is concerned with the topology of the phase space and the isoenergy manifold. We prove that the phase space and the isoenergy manifold of any billiard book are actually piecewise manifolds.