Show that if a one-parameter group of difeomorphisms of a symplectic manifold preserves the symplectic structure then it is a locally hamiltonian phase flow.
Note that
A locally hamiltonian vector field on a symplectic manifold $(M^{2n}, \omega^2)$ is the vector field $I \omega^1$, where $\omega^1$ is a closed 1-form on $M^{2n}$ and $\omega^1(\eta) = \omega^2(\eta, I\omega^1)$.