Dupin hypersurfaces and integrable hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants

Abstract

We establish a close relationship between hamiltonian systems of hydrodynamic type and hypersurfaces of a pseudoeuclidean space. This correspondence provides a complete classification of the 3 × 3 integrable nondiagonalizable hamiltonian systems, based upon the classification of Dupin hypersurfaces in E 4.