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 tau : 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.

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