ИСТИНА

A billiard's book is a new integrable Hamiltonian system that extends to the case of the billiard in a domain bounded by confocal quadrics. Such type of billiards formed by gluing a few classical billiard domains along pieces of their boundaries. The special case where we glue two domains called a topological billiard and was researched by V.V. Fokicheva. The billiard's book dynamical system has 4-dimensional phase space and two integrals: the scalar square of the velocity vector and one special integral. The second special integral can be described by the following feature of trajectories: the straight lines containing the segments of the polygonal billiard trajectory are tangents to a certain quadric (ellipse or hyperbola). Integrable Hamiltonian systems with 2 degrees of freedom have Fomenko-Zieschang invariants. Such invariants allow us to speak about the equivalence between closures of trajectories. Researching billiard's books we try model famous integrable systems in terms of Fomenko-Zieschang invariants. The Fomenko conjecture about modeling Fomenko-Zieschang invariants using billiard's books and new results that confirm the part of the conjecture will be presented.