Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

We study billiards on compact connected domains in $\mathbb{R}^3$ bounded by a finite number of confocal quadrics meeting in dihedral angles equal to ${\pi}/{2}$. Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence of the corresponding billiard systems on them. Billiard domains in $\mathbb{R}^3$ are classified with respect to combinatorial equivalence, resulting in 35 pairwise nonequivalent classes. For each of these obtained classes, we look for the homeomorphism class of the nonsingular isoenergy 5-manifold, and we show this to be one of three types: either $S^5$, or $S^1\times S^4$, or $S^2\times S^3$. We obtain 24 classes of pairwise nonequivalent (with respect to weak equivalence) Liouville foliations of billiards on these domains restricted to a nonsingular energy level. We also define bifurcation atoms of three-dimensional tori corresponding to the arcs of the bifurcation diagram. Bibliography: 59 titles.

billiard, integrable billiard, integrable system, Liouville foliation, topological invariants