We study a linear operator associated with a closed non‐exact 1‐form b defined on a smooth closed orientable surface M of genus . Here we present two proofs that reveal the interplay between the global solvability of the operator and the global topology of the surface. The first result brings an answer for the global solvability when the system is defined by a generic Morse 1‐form. Necessary conditions for the global solvability bearing on the sublevel and superlevel sets of primitives of a smooth 1‐form b have already been established; we also present a more intuitive proof of this result.
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Bergamasco et al. (2018) studied this question.