# About the projective Finsler metrizability: First steps in the non-isotropic case

## TL;DR

This paper explores the conditions under which certain second-order differential equations in Finsler geometry are metrizable by a Finsler metric, extending previous isotropic results to the non-isotropic case and identifying integrability conditions.

## Contribution

It advances the understanding of projective Finsler metrizability by analyzing the non-isotropic case and deriving new integrability conditions for the associated PDE system.

## Key findings

- Derived new integrability conditions for non-isotropic Finsler structures.
- Identified classes of non-isotropic sprays with integrable PDE systems.
- Extended the theory of Finsler metrizability beyond isotropic curvature cases.

## Abstract

We consider the projective Finsler metrizability problem: under what conditions the solutions of a given system of second-order ordinary differential equations (SODE) coincide with the geodesics of a Finsler metric, as oriented curves. SODEs with isotropic curvature have already been thoroughly studied in the literature and have proved to be projective Finsler metrizable. In this paper, we investigate the non-isotropic case and obtain new results by examining the integrability of the Rapcs\'ak system extended with curvature conditions. We consider the $n$-dimensional generic case, where the eigenvalues of the Jacobi tensor are pairwise different and compute the first and the higher order compatibility conditions of the system. We also consider the three-dimensional case, where we find a class of non-isotropic sprays for which the PDE system is integrable and, consequently, the corresponding SODEs are projective metrizable.