New classes of integrable billiards and their connection with invariants of integrable systems

Recently, the author has introduced a new class of integrable billiard books, which extends the well-known class of flat integrable billiards bounded by arcs of confocal quadrics.
"Billiard books" are glued from two-dimensional flat tables of integrable billiards (that is, bounded by confocal quadrics or concentric circles and their radii) along common arcs of the boundary. Permutations on the edges - the "spines" of the book - determine the transition of the ball from one sheet of the book to another after hitting the border of a flat piece. The design is well combined with the already known ones: a potential or a magnetic field is added.
The talk will focus on two subjects. First, these are the steps in proving A.T.Fomenko's conjecture on billiard modeling of integrable Hamiltonian systems. For an arbitrary non-degenerate singularity of an integrable system - a bifurcation of Liouville tori - we will show a billiard book that implements it. Such singularities-atoms correspond to the vertices of Fomenko-Zieschang molecules and define the bifurcation type of regular Liouville tori. In the second part of the talk, it is planned to highlight the topology of three-dimensional surfaces of constant energy for a number of systems of billiard books. The class of such manifolds turned out to be not limited to Seifert manifolds (as for integrable systems, Waldhausen manifolds were found, but not Seifert manifolds).