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A new four-dimensional family of skew-symmetric solutions of the Jacobiequations for Poisson structures is characterized. As a consequence, previouslyknown types of Poisson structures found in a diversity of physical situationsappear to be obtainable as particular cases of the new family of solutions.Additionally, it is possible to apply constructive methods for the explicitdetermination of fundamental properties of those solutions, such as theirCasimir invariants, symplectic structure and the algorithm for the reduction tothe Darboux canonical form, which have been reported only for a limited sampleof known finite-dimensional Poisson structures. Moreover, the results developedare valid globally in phase space, thus ameliorating the usual scope of Darbouxtheorem which is of local nature.

New four-dimensional solutions of the Jacobi equations for Poisson structures