On the global solvability of involutive systems

We consider a class of involutive systems in Tn+1Tn+1 associated with a closed 1-form defined on the torus TnTn. We prove that, under a geometric condition, the global solvability of this class is equivalent to a diophantine condition involving Liouville forms and the connectedness of all sublevel and superlevel sets of a global primitive associated with the system.