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