Classes of globally solvable involutive systems

We study a linear operator associated with a real smooth closed non-exact 1-form b defined on a closed orientable surface. Locally the operator can be seen as an overdetermined system of first order linear partial differential equations. Here we present a result that completely characterizes a class of systems that are globally solvable, namely when b has rank equal to 1, in terms of a topological condition. Such a condition bears on the superlevel and sublevel sets of primitives of b. In a certain covering space, called minimal covering space, the condition is equivalent to the connectedness of the superlevel and sublevel sets of the primitives there defined (a property that frequently appears in related papers). We furthermore exhibit another class of globally solvable systems by constructing smooth closed non-exact 1-forms of arbitrary rank on surfaces of genus greater than 1 out of 1-forms which individually define globally solvable systems on tori.