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

A systematic investigation of the skew-symmetric solutions of the three-dim ensional Jacobi equations is presented. As a result, three disjoint and com plementary new families of solutions are characterized. Such families are v ery general, thus unifying many different and well-known Poisson structures seemingly unrelated which now appear embraced as particular cases of a mor e general solution. This unification is not only conceptual but allows the development of algorithms for the explicit determination of important prope rties such as the symplectic structure, the Casimir invariants and the Darb oux canonical form, which are known only for a limited sample of Poisson st ructures. These common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework, instead of using a case-by-case approach. In addition, the metho ds developed are valid globally in phase space, thus ameliorating the usual scope of Darboux' reduction which is only of local nature. Finally, the fa milies of solutions found present some new nonlinear superposition principl es which are characterized. (C) 2001 American Institute of Physics.