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In this article we construct L--A representations of geodesic flows onquadrics and of billiard problems within ellipsoids in the pseudo--Euclideanspaces. A geometric interpretation of the integrability analogous to theclassical Chasles theorem for symmetric ellipsoids is given. We also consider ageneralization of the billiard within arbitrary quadric allowing virtualbilliard reflections.

Geodesic and billiard flows on quadrics in pseudo-Euclidean spaces: L-A pairs and Chasles theorem