Realization of numeriсal invariant of the Siefert bundle of integrable systems by billiards

A local case of A. Fomenko conjecture on possibility of realization of a Liouville foliation with arbitrary topological Fomenko–Zieschang invariant (which is a graph with numerical marks) is discussed. In the class of billiard books, a foliation with arbitrary value of one integer mark (that corresponds to Euler class of one Seifert submanifold) was realized.