Reciprocal transformations and flat metrics on Hurwitz spaces

We consider hydrodynamic systems which possess a local Hamiltonian structure. To such a system, there are also associated an infinite number of nonlocal Hamiltonian structures. We give the necessary and sufficient conditions so that, after a nonlinear transformation of the independent variables, the reciprocal system still possesses a local Hamiltonian structure. We show that, under our hypotheses, bi-Hamiltonicity is preserved by the reciprocal transformation. Finally, we apply such results to the reciprocal systems of genus g Whitham-KdV modulation equations.