About the integrability of the Rapcs\'ak equation

In [17] A. Rapcs\'ak obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential equations. In this paper we investigate the integrability of the Rapcs\'ak system, consisting of the Rapcs\'ak equations and the homogeneity condition, by using the Spencer version of the Cartan-K\"ahler theorem. We also consider the extended Rapcs\'ak system, where the first integrability conditions are included.