  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.
