Commutative conservation laws for geodesic flows of metric admitting projective symmetry (Q1599945)

Suppose that a pseudo-Riemannian manifold \((M,g)\) admits a vector field \(X\) whose one-parameter group maps any geodesic line on some geodesic line (up to parametrization). Let \(r\) denote the maximal value on \(M\) of the degree of the minimal polynomial of \(((g^{ik} ({\mathcal L}_xg)_{kj}))_{ij}\). Then \(r\) explicit independent quadratic forms on \(TM\) are exhibited that are first integrals of the geodesic flow of \((M,g)\). This gives new first integrals of the geodesic flow, once a vector field preserving geodesic lines is known.