Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures

We present a number of alternative sufficient conditions for the existence of pre-symplectic or pre-implectic (Poisson) structures, for both ordinary differential (ODE) and ordinary difference (OΔE) equations. Four alternative sets of conditions are obtained for ODEs and OΔEs in n dependent variables: (1) A vector field in involution with the ODE and an integral (or two symmetries for an OΔE) imply a pre-implectic structure; (2) volume preservation and n−2 symmetries imply a pre-symplectic structure; (3) volume preservation and n−2 integrals imply a pre-implectic structure; (4) complex implectic structure implies infinitely many real implectic structures. In all but the first case the methods can give a set of distinct structures.