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We show that under certain technical assumptions any weakly nonlocalHamiltonian structure compatible with a given nondegenerate weakly nonlocalsymplectic structure $J$ can be written as the Lie derivative of $J^{-1}$ alonga suitably chosen nonlocal vector field. Moreover, we present a new descriptionfor local Hamiltonian structures of arbitrary order compatible with a givennondegenerate local Hamiltonian structure of zero or first order, includingHamiltonian operators of the Dubrovin-Novikov type.

Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility