On the projective Finsler metrizability and the integrability of Rapcsak equation

A. Rapcsak obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsak system and the extended Rapcsak system, by using the Spencer version of the Cartan-Kahler theorem. We also consider the extended Rapcsak system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.