ИСТИНА

A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflection from a boundary. We used Fomenko's theory [1] to describe topology of joint integrable surfaces. In this report we are going to discuss billiards with nonconvex angles bounded by arcs of several confocal conics [2]. The integral trajectory of the flow going into a singular point can not be extend for the all values of the parameter (time). In contrast to the classical case of complete flows, the regular leaves of the Liouville foliation are the spheres with handles and punctures, rather than Liouville tori. Thm. (V.Moskvin, 2018) Consider billiard Ω on the plane. Let Σ1, . . . , ΣN be a full partition of billiard Ω. 2-dimmensional singular fiber of 3-dimmensional atom can be algorithmically constructed of (4N) disks I × I and (n) 1-dimmensional graphs, here n ≤ 4N − 1. This work was supported by the program “Leading Scientific Schools” (grant no. NSh-6399.2018.1, Agreement No. 075-02-2018-867) [1] A.V. Bolsinov and A.T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification., Chapman/ Hall/CRC, Boca Raton, 2004. [2] V. Dragovic, M. Radnovic, Pseudo-integrable billiards and arithmetic dynamics, Journal of Modern Dynamics, Volume 8, No. 1, 2014, 109-132 p.