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We review properties of so-called special conformal Killing tensors on aRiemannian manifold $(Q,g)$ and the way they give rise to a Poisson-Nijenhuisstructure on the tangent bundle $TQ$. We then address the question ofgeneralizing this concept to a Finsler space, where the metric tensor fieldcomes from a regular Lagrangian function $E$, homogeneous of degree two in thefibre coordinates on $TQ$. It is shown that when a symmetric type (1,1) tensorfield $K$ along the tangent bundle projection $\tau: TQ\to Q$ satisfies adifferential condition which is similar to the defining relation of specialconformal Killing tensors, there exists a direct recursive scheme again forfirst integrals of the geodesic spray. Involutivity of such integrals,unfortunately, remains an open problem.

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