The generalised hodograph method for non-diagonalisable integrable systems of hydrodynamic type

We extend the generalised hodograph method to non-diagonalisable integrable systems of hydrodynamic type coming from F-manifolds. The method allows the construction of solutions starting from the symmetries of the system. In the diagonal case, the completeness of the symmetries follows from the integrability conditions that ensure the applicability of a Darboux’s theorem on Pfaffian systems. In the regular non-diagonalisable case the validity of this theorem relies on some further assumptions that we discuss in detail. Under these assumptions, the method provides the general solution as in Tsarev’s diagonal case.