  We describe an averaging procedure on a Dirac manifold, with respect to a
class of compatible actions of a compact Lie group. Some averaging theorems on
the existence of invariant realizations of Poisson structures around (singular)
symplectic leaves are derived. We show that the construction of coupling Dirac
structures (invariant with respect to locally Hamiltonian group actions) on a
Poisson foliation is related with a special class of exact gauge
transformations.
