Symmetric Hamiltonian vector fields

In this work, we present algebraic results for Hamiltonian and symmetric vector fields on 2n-dimensional symplectic vector spaces. Our main results are in the linear context. In the case n = 1, we have that a linear vector field is Hamiltonian if and only if it is reversible by an involution. For n>1 and under certain conditions, we obtain a similar correspondence in which the symmetry group is generated by the symplectic form and the reversing symmetries are not necessarily involutions. In the non-linear context, we exhibit families of vector fields that are Hamiltonian with symmetry group Z4.