Integrability of a geodesic flow on the intersection of several confocal quadrics

The classical Jacobi–Chasles theorem states that tangent lines drawn at all points of a geodesic curve on a quadric in $n$-dimensional Euclidean space are tangent, in addition to the given quadric, to $n–2$ other confocal quadrics, which are the same for all points of the geodesic curve. This theorem immediately implies the integrability of a geodesic flow on an ellipsoid. In this paper, we prove a generalization of this result for a geodesic flow on the intersection of several confocal quadrics. Moreover, if we add the Hooke’s potential field centered at the origin to such a system, the integrability of the problem is preserved.