Transformations of quasilinear systems originating from the projective theory of congruences

Abstract

We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the L\'evy transformations of semihamiltonian systems. Correspondence between commuting flows and certain families of planes containing the lines of the congruence is pointed out. In the particular case n=2 this construction provides an explicit parametrization of surfaces, harmonic to a given congruence. Adjoint L\'evy transformations of semihamiltonian systems are discussed. Explicit formulae for the L\'evy and adjoint L\'evy transformations of the characteristic velocities are set down. A closely related construction of the Ribaucour congruences of spheres is discussed in the Appendix.