Non-homogeneous 1+0 Hamiltonian operators in Darboux form are characterized by a Lie algebra, a non-degenerate quadratic Casimir, and a 2-cocycle, with an explicit catalogue of all cases through dimension 6.
and Winternitz, P.: Classification and Identification of Lie Al- gebras
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Lie algebras with compatible scalar products for non-homogeneous Hamiltonian operators
Non-homogeneous 1+0 Hamiltonian operators in Darboux form are characterized by a Lie algebra, a non-degenerate quadratic Casimir, and a 2-cocycle, with an explicit catalogue of all cases through dimension 6.