  In this paper, we focus on the construction of high order volume preserving
in- tegrators for divergence-free vector fields: the monomial basis, the
exponential basis and tensor product of the monomial and the exponential basis.
We first prove that the commutators of elementary divergence-free vector fields
(EDFVF) of those three kinds are still divergence-free vector fields of the
same kind. Assuming then there is only diagonal part of divergence-free vector
field of the monomial basis, for those three kinds of divergence-free vector
fields, we construct high order volume-preserving inte- grators using the
multi-commutators for EDFVFs. Moreover, we consider the ordering of the EDFVFs
and their commutators to reduce the error of the schemes, showing by numerical
tests that the strategy in [9] works very well.
