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.