  We study analytic deformations of holomorphic differential 1-forms. The
initial 1-form is exact homogeneous and the deformation is by polynomial
integrable 1-forms. We investigate under which conditions the elements of the
deformation are still exact or, more generally, exhibit a first integral. Our
results are related to natural extensions of classical results of Ilyashenko on
limit cycles of perturbations of hamiltonian systems in two complex variables.
