  In this paper we investigate the following question: under what conditions
can a second-order homogeneous ordinary differential equation (spray) be the
geodesic equation of a Finsler space. We show that the Euler-Lagrange partial
differential system on the energy function can be reduced to a first order
system on this same function. In this way we are able to give effective
necessary and sufficient conditions for the local existence of a such Finsler
metric in terms of the holonomy algebra generated by horizontal vector-fields.
We also consider the Landsberg metrizability problem and prove similar results.
This reduction is a significant step in solving the problem whether or not
there exists a non-Berwald Landsberg space.
