Gel'fand-Dorfman theorem and exact cocycle Poisson structures (Q1891487)

A starting point here is the Gel'fand-Dorfman theorem on functions in involution on bi-Hamiltonian manifolds [see \textit{I. M. Gel'fand} and \textit{I. Ya. Dorfman}, Funct. Anal. Appl. 13, 248-262 (1980); translation from Funkts. Anal. Prilozh. 13, No. 4, 13-30 (1979; Zbl 0428.58009)]. The authors give a generalization of the theorem to Poisson manifolds which (unlike the one of Gel'fand-Dorfman [\textit{I. M. Gel'fand} and \textit{I. Ya. Dorfman}, Funct. Anal. Appl. 14, 223-226 (1981); translation from Funkts. Anal. Prilozh. 14, No. 3, 71-74 (1980; Zbl 0444.58010)]) uses some cohomological conditions. Some conditions to construct compatible Poisson structures (exact cocycle type) are found. It is shown that every Lie- Poisson structure is exact with the dilation vector on the manifold.