In this paper, we study the vector fields X with a global Poincaré cross-section on a 2n + 1-dimensional presymplectic manifold (M, ῶ) under certain conditions. We use (M, ῶ) to construct a 2n + 2k dimensional symplectic manifold ( \({\tilde M},\ \Lambda\) ), on which the vector field X can be extended to a Hamiltonian vector field \({\tilde X}\) with a smooth Hamiltonian \(H:{\tilde M}\rightarrow R\) . We also consider vector fields X with a first integral F and a Jacobi multiplier J on an n-dimensional manifold (M, Ω). On a level set Σ of F, we get an n − 1-volume form ωn on Σ and prove that X is a volume-preserving vector field with respect to ωn. Specifically, when X is a 3 dimensional devergence-free vector field, the results have been discussed by Lerman in 2019.
