  We prove that a local Hamiltonian operator of hydrodynamic type K_1 is
compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type
K_2 if and only if the operator K_1 is locally the Lie derivative of the
operator K_2 along a vector field in the corresponding domain of local
coordinates. This result gives a natural invariant definition of the class of
special flat manifolds corresponding to all the class of compatible
Dubrovin--Novikov Hamiltonian operators (the Frobenius--Dubrovin manifolds
naturally belong to this class). There is an integrable bi-Hamiltonian
hierarchy corresponding to every flat manifold of this class. The integrable
systems are also studied in the present paper. This class of integrable systems
is explicitly given by solutions of the nonlinear system of equations, which is
integrated by the method of inverse scattering problem.
