Modeling bifurcations of Liouville foliations of integrable Hamiltonian systems by billiard books

Consider a two-dimensional cell complex whose two-dimensional cells are flat billiards bounded by arcs of confocal quadrics. We enumerate all two-dimensional cells and assign to each one-dimensional edge of the complex - the so-called spine of the book - a cyclic permutation of the numbers of sheets adjacent to this edge. Project all billiard sheets isometrically onto a plane. If the edges of the complex under this projection fall into one arc of the quadric, then we combine the cycles assigned to them into one permutation. We require that the permutations assigned to two arcs of quadrics that have a common point commute with each other. We call such a two-dimensional complex with assigned permutations a billiard book. The construction of billiard books made it possible not only to significantly extend the class of integrable billiard systems, but also to discover new Liouville foliations. These Liouville foliations (encoded by the Fomenko-Zieschang invariants), on the one hand, have not previously been encountered in classical problems of dynamics, and on the other hand, the billiard systems corresponding to them have a visual description. In this regard, A.T.Fomenko formulated a program conjecture on the realizability by integrable billiards of arbitrary Liouville foliations (i.e. labeled molecules) of nondegenerate integrable systems with two degrees of freedom (in the class of Liouville equivalence). In particular, it was proved that the Liouville foliations of billiard books contain all bifurcations of the Liouville tori of non-degenerate Hamiltonian systems, as well as arbitrary bases of Liouville foliations.