  In earlier work, Lomeli and Meiss used a generalization of the symplectic
approach to study volume preserving generating differential forms. In
particular, for the $\mathbb{R}^3$ case, the first to differ from the
symplectic case, they derived thirty-six one-forms that generate exact volume
preserving maps. Xue and Zanna had studied these differential forms in
connection with the numerical solution of divergence-free differential
equations: can such forms be used to devise new volume preserving integrators
or to further understand existing ones? As a partial answer to this question,
Xue and Zanna showed how six of the generating volume form were naturally
associated to consistent, first order, volume preserving numerical integrators.
In this paper, we investigate and classify the remaining cases. The main result
is the reduction of the thirty-six cases to five essentially different cases,
up to variable relabeling and adjunction. We classify these five cases,
identifying two novel classes and associating the other three to volume
preserving vector fields under a Hamiltonian or Lagrangian representation. We
demonstrate how these generating form lead to consistent volume preserving
schemes for volume preserving vector fields in $\mathbb{R}^3$.
