  In this work we investigate the set of cubic Hamiltonian vector fields for
which their associated Kahan-Hirota-Kimura maps preserve the original
Hamiltonian function. We analyze these fields in $\mathbb{R}^2$ and
$\mathbb{R}^4$. We also study a family of fields in $\mathbb{R}^6$.
Additionally, we explore several properties like the existence of additional
first integrals of specific type, the possibility that the initial Hamiltonian
vector field is a Lie symmetry of the corresponding map, or the symplecticity
of the considered maps.
