Double waves in multi-dimensional systems of hydrodynamic type: the necessary condition for integrability

The invariant differential-geometric approach to the integrability of (2+1)- dimensional systems of hydrodynamic type is developed. It is argued that the existence of special solutions known as `double waves' is generically equivalent to the diagonalizability of an arbitrary matrix of some two-parameter family of matrices associated with the system. Since the diagonalizability can be effectively verified by differential-geometric means, this provides a simple necessary condition for integrability.