Integrable Mechanical Billiards in Higher-Dimensional Space Forms

In this article, we consider mechanical billiard systems defined with Lagrange's integrable {extension} of Euler's two-center problems in the Euclidean space, the sphere, and the hyperbolic space of arbitrary dimension $n \geqslant 3$. In the three-dimensional Euclidean space, we show that the billiard systems with any finite combinations of {spheroids and circular hyperboloids of two sheets} having two foci at the Kepler centers are integrable. The same holds for the projections of these systems on the three-dimensional sphere and in the three-dimensional hyperbolic space by means of central projection. Using the same approach, we also extend these results to the $n$-dimensional cases.