Towards the physical significance of the $(k^2+A)\mathrm{\parallel }u\mathrm{\parallel }$ metric

We offer an example of the second order Kawaguchi metric function the extremal flow of which generalizes the flat space-time model of the semi-classical spinning particle to the framework of the pseudo-Riemannian space-time. The general shape of the variational Euler-Poisson equation of the fourth order in the (pseudo-)Riemannian space is being developed too.