Billiard books of low complexity and realization of Liouville foliations of integrable systems

In paper we study the topology of integrable billiard books, (i.e. systems on CW-complexes glued from flat domains of confocal billiards. Significant progress has been made in proving the local version of the billiard Fomenko conjecture. In particular, billiards were used to realize an important class of subgraphs of the Fomenko - Zieschang graph invariants (that classify Liouville foliations of integrable systems in topological sense). Then we classify in combinatorial sense billiard books of low complexity (with a small number of one-dimensional cells), glued from flat domains that contain foci of the family of quadrics. Calculation of Fomenko–Zieschang invariants for these systems is in progress.