Metrizability of holonomy invariant projective deformation of sprays

In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions: Starting by a Finsler spray $S$ and a holonomy invariant function $P$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\lambda P C$. We prove that for any holonomy invariant nontrivial function $P$ and for almost every value $\lambda\in R$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray: in these cases, the holonomy invariant function is necessarily one of the principal curvatures of the geodesic structure.