Dynamical systems and Poisson structures

We first consider the Hamiltonian formulation of $n=3$ systems, in general, and show that all dynamical systems in $R3$ are locally bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. The construction of the Poisson structures is based on solving an associated first order linear partial differential equations. We find the Poisson structures of a dynamical system recently given by Bender et al [J. Phys. A: Math. Theor. 40, F793 (2007)]. Secondly, we show that all dynamical systems in $Rn$ are locally $(n−1)$-Hamiltonian. We give also an algorithm, similar to the case in $R3$⁠, to construct a rank two Poisson structure of dynamical systems in $Rn$⁠. We give a classification of the dynamical systems with respect to the invariant functions of the vector field $X⃗$ and show that all autonomous dynamical systems in $Rn$ are superintegrable.