Global properties of a class of overdetermined systems

We consider an overdetermined system of complex vector fields on the three-dimensional torus which is naturally associated to a real analytic, closed 1-form on the two-dimensional torus. By means of a detailed study of the geometry of level sets of a primitive of the pull-back of the 1-form via the universal covering, we prove that a necessary condition for the system to be globally solvable, in the non-exact case, is that each connected component of the critical set has a point at which the local primitives of the 1-form are open maps. When this condition is violated we also construct global solutions to the inhomogenous equations having analytic singularities.