  In this paper we introduce a new class of integrable systems, naturally
associated to Hurwitz spaces (spaces of meromorphic functions over Riemann
surfaces). The critical values of the meromorphic functions play the role of
"times". Our systems give a natural generalization of the Ernst equation; in
genus zero they realize the scheme of deformation of integrable systems
proposed by Burtsev, Mikhailov and Zakharov.
  We show that any solution of these systems in rank 1 defines a flat diagonal
metric (Darboux-Egoroff metric) together with a class of corresponding systems
of hydrodynamic type and their solutions.
