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

1 (Dubrovin–Novikov Hamiltonian operator [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, first of all, a convenient general invariant criterion of the compatibility for the Dubrovin–Novikov Hamiltonian operators and, in addition, this 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 of flat manifolds). 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. The corresponding results on compatible nonlocal Hamiltonian operators of hydrodynamic type and other Hamiltonian and symplectic differential-geometric type operators related by the Lie derivative and the results on the corresponding to them integrable bi-Hamiltonian systems will be published in other our works. Recall that an operator K [u(x)] is called Hamiltonian if it defines a Poisson bracket (skew-symmetric and satisfying the Jacobi identity)