Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems

We consider a generalization of a mathematical billiard bounded by arcs of confocal quadrics, known as billiard books. Billiard books define a large class of integrable Hamiltonian systems. In this connection the question arises of the possibility of realizing integrable Hamiltonian systems with two degrees of freedom by billiard books. The authors have proved previously that for any nondegenerate three-dimensional bifurcation ($3$-atom) a billiard book in which such a bifurcation appears can be constructed algorithmically. Based on the preceding result, we give a proof of the fact that given any base of a Liouville foliation (rough molecule), a billiard book can be constructed algorithmically such that the base of the Liouville foliation of this system is isomorphic to the one given initially.
Bibliography: 15 titles.