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The aim of this paper is to consider quasi-linear systems which are not inthe form of evolution equations. We propose new condition of Richness orSemi-Hamiltonicity for such a system and prove that the blow up analysis alongcharacteristic curves can be performed for it in an analogous manner. Thisopens a possibility to use this ansatz also for geometric problems. We applythe results to the problem of Polynomial integral for geodesic flows on the2-torus.

Richness or Semi-Hamiltonicity of quasi-linear systems which are not in evolution form