Finsler metrizabilities and geodesic invariance

We demonstrate that various metrizability problems for Finsler sprays can be reformulated in terms of the geodesic invariance of two tensors, namely the metric and angular tensors. We show that a spray $S$ is the geodesic spray of some Finsler metric if and only if its metric tensor is geodesically invariant. Moreover, we establish that gyroscopic sprays constitute the largest class of sprays characterized by a geodesic-invariant angular metric. Scalar functions associated with these geodesically invariant tensors will also be invariant, thereby providing first integrals for the given spray.