Hamiltonian equations in R 3

The Hamiltonian formulation of N=3 systems is considered in general. The most general solution of the Jacobi equation in R 3 is proposed. The form of the solution is shown to be valid also in the neighborhood of some irregular points. Compatible Poisson structures and corresponding bi-Hamiltonian systems are also discussed. Hamiltonian structures, the classification of irregular points and the corresponding reduced first order differential equations of several examples are given.