A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold⋆

We review properties of so-called special conformal Killing tensors on a Riemannian manifold (Q, g) and the way they give rise to a Poisson–Nijenhuis structure on the tangent bundle TQ. We then address the question of generalizing this concept to a Finsler space, where the metric tensor field comes from a regular Lagrangian function E, homogeneous of degree two in the fibre coordinates on TQ. It is shown that when a symmetric type (1,1) tensor field K along the tangent bundle projection τ : TQ → Q satisfies a differential condition which is similar to the defining relation of special conformal Killing tensors, there exists a direct recursive scheme again for first integrals of the geodesic spray. Involutivity of such integrals, unfortunately, remains an open problem.