INTEGRABILITY OF A GEODESIC FLOW ON THE INTERSECTION OF SEVERAL CONFOCAL QUADRICS

The classical Jacobi-Schall theorem states that Tangent lines drawn at all points of a geodesic curve on a quadric in n-dimensional Euclidean space are tangent, as well as 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 of coordinates to such a system, the integrability of the problem is preserved.