The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability

An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type, is developed. We prove that the existence of special solutions known as ‘double waves’ is equivalent to the diagonalizability of an arbitrary matrix of the two-parameter family Since the diagonalizability can be effectively verified by calculating the Haantjes tensor, this provides a simple necessary condition for integrability.