ИСТИНА

Let us recall that billiard system describes motion of a particle in a flat domain Q with piecewise smooth boundary P. Reflection should be elastic. The Hamiltonian is the square of velocity vector. D.Birkhoff proved the integrability if P is an ellipse. V.V.Kozlov, D.V.Treschev proved that integrability preserves for P that consists of arcs of confocal ellipses and hyperbolas. This system has an additional first integral A which value is some parameter of the caustic for trajectory. Fixing |"v|2 = h one have 3-dimensional manifold 03 foliated on level surfaces of A. Such foliations are smooth-wise analogs of Liouville foliations investigated by A.T. Fomenko school. Fomenko-Zieschang invariant (graph with numerical marks which vertices correspond to singularities of the foliation) classifies them in the sense of Liouville equivalence. Two integrable systems are equivalent if piece-wise diffeomorphism exists. Their trajectory closures also have the same structure. Fomenko-Zieschang invariants of two systems coincides iff they are Liouville equivalent. Fomenko-Zieschang invariant for the plane billiards were calculated by V. Dragovich, M. Rad-novich and V.V. Vedyushkina (Fokicheva). Such plane domains we call elementary domains. By gluing together elementary domains-sheets along common borders, it is possible to produce complex topological billiards (in the case that no more than two sheets are glued along a common border). An interesting problem is to describe the Liouville foliation of the obtained billiards. In terms of the Fomenko-Zieschang invariant, the author answered for all topological billiards. New class of integrable billiard was constructed by gluing together several flat domains by some common boundary arcs of their immersions in the plane. This class of billiards was called ’’billiard's books". Note that if number of ’’sheets’’ is more than 2, some permutation should correspond to the gluing arc. This permutation shows locally the next sheet for trajectory reaching this boundary. We will talk about integrable billiards that realize one up-to-date way of modeling integrable Hamiltonian systems (including classical mechanical cases of integrability and geodesic flows on 2-dimensional surfaces) and its singularities. Roughly speaking, the ’’complexity” of first integrals of initial system or flow corresponds to the complexity of billiard domain that has a structure of two-dimensional CW-complex. The Hamiltonian and first integral of such billiard system are quadratic.