Given a volume form on an oriented Poisson manifold one can define the curl (or modular) vector field with respect to this form. Under some nondegeneracy conditions, curl vector fields have been used to "normalize" their parent Poisson structure and classify classes of Poisson structures (J.P. Dufour, A. Haraki and A. Wade). A Poisson structure P is said to be exact (or unimodular) if one of its curl vector fields is trivial. What can we say about such a Poisson structure? Can it still be "normalized"? In this talk I will concentrate on the 4-dimensional case, and I will show how to produce normal forms for two classes of exact Poisson structures. This is joint work with H. Mena Matos.