Planar vector fields in the kernel of a 1--form

We classify, up to a natural equivalence relation, vector fields of the plane which belong to the kernel of a 1--form. This form can be closed, in which case the vector fields are integrable, or not, in which case the differential of the form defines a, possibly singular, symplectic form. In every case, we provide a fairly complete list of local models for such fields and construct their transversal unfoldings. Thus, the local bifurcations of vector fields of interest can be studied, among them being the integrable fields of the plane.