  In this paper we characterize sprays that are metrizable by Finsler functions
of constant flag curvature. By solving a particular case of the Finsler
metrizability problem we provide the necessary and sufficient conditions that
can be used to decide whether or not a given homogeneous system of second order
ordinary differential equations represents the Euler-Lagrange equations of a
Finsler function of constant flag curvature. The conditions we provide are
tensorial equations on the Jacobi endomorphism. We identify the class of
homogeneous SODE where the Finsler metrizability is equivalent with the
metrizability by a Finsler function of constant flag curvature.
