Topology of $5$-surfaces of a 3D billiard inside a triaxial ellipsoid with Hooke's potential

A billiard inside a triaxial ellipsoid in a Hooke potential field (both attractive and repulsive) is considered. For each zone of non-bifurcational values of the energy, the homeomorphism class of the corresponding isoenergy $5$-surface in the phase space is determined. This result was obtained without using the integrability of the system. Following the method of V. V. Kozlov, we also present an explicit form of $n$ involutive first integrals for the multidimensional generalization of studied problem, i.e., a billiard in a Hooke potential field inside an $n$-axial ellipsoid in $n$-dimensional space.