Richness or semi-Hamiltonicity of quasi-linear systems that are not in evolution form

The aim of this paper is to consider strictly hyperbolic quasi-linear systems of conservation laws which appear in the form $A(u)u_{x}+B(u)u_{y}=0.$ If one of the matrices $A(u), B(u)$ is invertible, then this system is in fact in the form of evolution equations. However, it may happen that traveling along characteristics one moves from the “chart” where $A(u)$ is invertible to another “chart” where $B(u)$ is invertible. We propose a new condition of richness or semi-Hamiltonicity for such a system that is “chart”-independent. This new condition enables one to perform the blow-up analysis along characteristic curves for all times, not passing from one “chart” to another. This opens a possibility to use this theory for geometric problems as well as for stationary solutions of 2D+1 systems. We apply the results to the problem of polynomial integral for geodesic flows on the 2-torus.