Generalizations of Jacobi-Chasles and Graves Theorems

The most illustrative integrable Hamiltonian systems are billiards bounded by confocal quadrics. The integrability of these systems follows from the classical Jacobi–Chasles theorem. Recall that according to this theorem, \textit{tangent lines drawn to a geodesic on an $n$-axial ellipsoid in Euclidean $\mathbb{R}^n$ touch, in addition to this ellipsoid, $n-2$ confocal quadrics common to all points of the given geodesic}. This theorem implies the integrability of the geodesic flow on the ellipsoid.
V.A. Kibkalo investigated the issue of integrability of the geodesic flow on the intersection of several confocal quadrics. He showed that the geodesic flow on the intersection of $(n-2)$ confocal quadrics is a completely integrable Hamiltonian system. It turns out that the result remains valid if we consider the geodesic flow on the intersection of an arbitrary number of non-degenerate confocal quadrics. Moreover, the following theorem holds.

\begin{theorem}[Belozerov] Let </nomathmode><mathmode>$Q_1, \ldots, Q_k$ be non-degenerate confocal quadrics of different types in $\mathbb{R}^n$ and $Q = \bigcap_{i=1}^k Q_i$. Then:

• the geodesic flow on $Q$ is quadratically integrable;
• tangent lines drawn to all points of a geodesic on $Q$, in addition to $Q_1, \ldots, Q_k$, touch $n-k-1$ quadrics confocal with $Q_1, \ldots, Q_k$ and common to all points of this geodesic.

\end{theorem}
</mathmode><nomathmode>
Remark. Geodesics on the intersection of non-degenerate confocal quadrics, in general, are not geodesics on any of the quadrics $Q_1, \ldots, Q_k$. Therefore, Theorem 1 is not a consequence of the classical Jacobi-Chasles theorem.
According to Theorem 1 and the result of V.V. Kozlov on integrable geodesic flows on two-dimensional surfaces, the connected component of the compact intersection of $(n-2)$ quadrics is homeomorphic either to a torus $T^2$ or to a sphere $S^2$. Both cases are realized. Nevertheless, it is possible to describe the class of homeomorphism of any compact intersection of non-degenerate confocal quadrics. It turns out that it is homeomorphic to a direct product of spheres.
It also turned out that the classical Jacobi-Chasles theorem can be generalized not only for Euclidean spaces, but also for pseudo-Euclidean spaces and spaces of constant curvature. These generalizations significantly enrich the class of integrable billiards.
Studying the trajectory properties of multidimensional billiards bounded by ellipsoids, the author and his scientific advisor A.T. Fomenko obtained a generalization of two more classical results — the focal property of quadrics and Graves' theorem. Recall that the classical Graves' theorem states that if you put an inextensible loop on an ellipse and, stretching the thread with a pencil to the limit, draw a curve, the result will be an ellipse confocal with the given one. It turns out that this fact has a multidimensional generalization, so ellipsoids of arbitrary dimension can be constructed using a thread.