Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative, and integrable systems of hydrodynamic type

We prove that two Dubrovin-Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the class of flat manifolds, which correspond to arbitrary, pairs of compatible Dubrovin-Novikov Hamiltonian operators. Locally, these manifolds are defined by solutions of a system of nonlinear equations, which is integrable by the method of the inverse scattering problem. We construct the integrable hierarchies generated by arbitrary, pairs of compatible Dubrovin-Novikov Hamiltonian operators.