  We study groups of formal or germs of analytic diffeomorphisms in several
complex variables. Such groups are related to the study of the transverse
structure and dynamics of Holomorphic foliations, via the notion of holonomy
group of a leaf of a foliation. For dimension one, there is a well-established
dictionary relating analytic/formal classification of the group, with its
algebraic properties (finiteness, commutativity, solvability, ...). Such system
of equivalences also characterizes the existence of suitable {\it integrating
factors}, i.e., invariant vector fields and one-forms associated to the group.
In this paper we search the basic lines of such dictionary for the case of
several complex variables groups. For abelian, metabelian, solvable or
nilpotent groups we investigate the existence of suitable formal vector fields
and closed differential forms which exhibit an invariance property under the
group action. Our results are applicable in the construction of suitable
integrating factors for holomorphic foliations with singularities. We believe
they are a starting point in the study of the connection between Liouvillian
integration and transverse structures of holomorphic foliations with
singularities in the case of arbitrary codimension.
