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On the global solvability for overdetermined systems. (English) Zbl 1275.35004
Summary: We consider a class of systems of two smooth vector fields on the 3-torus associated to a closed 1-form. We prove that the global solvability is completely determined by the connectedness of the sublevel and superlevel sets of a primitive of this 1-form in the minimal covering.

MSC:
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35N10 Overdetermined systems of PDEs with variable coefficients
58J10 Differential complexes
35B10 Periodic solutions to PDEs
35F05 Linear first-order PDEs
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