Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type, and \(N\)-wave systems (Q1917360)

The author investigates low-dimensional homogeneous isoparametric hypersurfaces in connection with a Hamiltonian system on the ambient Euclidean space. Let \(M\) be an isoparametric hypersurface with \(g\) distinct principal curvatures and multiplicities \(m_1\), \(m_2\). For the cases \((g,m_1,m_2)=(3,1,1), (3,2,2), (4,1,1), (4,1,2), (4,1,3), (4,2,2), (6,1,1)\) the author gives explicit descriptions of the Cartan-Münzner polynomials. The corresponding symmetric space is then used to show that the Hamiltionian system is integrable, but not diagonalizable. Further possible generalizations are pointed out. The paper is clearly written and contains 58 references.