ИСТИНА

Consider an integrable billiard in a planar domain bounded by arcs of confocal ellipses and hyperbolas. Let us consider a domain bounded by arcs of the focal line, an ellipse and two hyperbolas. Glue n copies of such billiard along two convex boundary arcs and the segment of the focal line. Let us define the motion as follows: when hitting a boundary arc, the billard ball will change the domain by some permutation. Using this construction (which is called billiard book) we can model interesting effects in the topology of the foliations on the isoenergy 3-dimensional manifolds of integrable Hamiltonian systems. The first there exist permutations that, being attributed to the convex arcs and the segment of the focal straight line, make it possible to obtain a foliation in the neighborhood of an unstable trajectory any non-degenerate saddle bifurcation of Liouville tori. The second, if one of this permutation is cyclic permutation and the permutation on the adjacent arc is the some degree k of the given cyclic permutation then the isoenergy 3-manifold is homeomorphic to the lens space L(n,k). This lens space is stratified into levels of the additional first integral which is the parameter of the confocal quadric (called caustic) tangent to all straight lines containing the links of the billiard path. As in the planar case, the motion along convex boundary arcs is stable, and the motion along the focal line segment is unstable. Moreover we show the constructions of the integrable billiards which model the isoenergy manifolds of the typical integrable case of the rigid body dynamics.