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In the paper we discuss Fomenko conjecture on realization of topology oftopology of Liouville foliaions of smooth and real-analytic integrableHamiltonian systems by integrable billiards. Vedyushkina-Kharcheva algorithm of3-atom realization by billiard books is described clearly in the terms of$f$-graphs. Note, that an arbitrary type of base of Liouville foliation on thewhole isoenergy surface was also realized by V. Vedyushkina and I. Kharchevaalgorithm. In this paper this algorithm is visually applied to realization of awell-known Zhukovskii integrable system (Euler system with a gyrostat) in acertain energy zone. It turns out that Liouville foliation is also realizedusing this construction, not only the class of its base. Thus, Liouvilleequivalence of these billiard and mechanical systems is shown. Then we discussV. Kibkalo and V. Vedyushkina result on constructing of a billiard with anarbitrary numerical invariant. Existence of Fomenko-Zieschang invariant isproved for some subclass of billiard books without potential. Also, it isproved that such systems are topologically stable. In the end of the paper weshow that planar billiard system in some potential field can have anon-degenerate rank 1 4-singularity singularity not satisfying non-splittingcondition.  Keywords: mathematical billiard, billiard book, integrable Hamiltoniansystem, rigid body dynamics, Liouville foliation, Fomenko-Zieschang invariant,singularity.

Realization of Integrable Hamiltonian Systems by Billiard Books