Integrable four-component systems of conservation laws and linear congruences in ${\mathbb P}^5$

We propose a differential-geometric classification of the fourcomponent hyperbolic systems of conservation laws which satisfy the following properties: (a) they do not possess Riemann invariants; (b) they are linearly degenerate; (c) their rarefaction curves are rectilinear; (d) the cross-ratio of the four characteristic speeds is harmonic. This turns out to provide a classification of projective congruences in ${\mathbb P}^5$ whose developable surfaces are planar pencils of lines, each of these lines cutting the focal variety at points forming a harmonic quadruplet. Symmetry properties and the connection of these congruences to Cartan’s isoparametric hypersurfaces are discussed.