The main purpose of this paper is to show, in the two-dimensional torus, a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s-solvable. We are also interested in studying its global s-hypoellipticity. We present connections between these global concepts and a priori estimates. We also present two applications of our results for systems of operators with variable coefficients.

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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.

We show that almost all perturbations P — λ, λ € C, of an arbitrary constant coefficient partial differential operator P are globally hypoelliptic on the torus. We also give a characterization of the values λ € C for which the operator is globally hypoelliptic; in particular, we show that the addition of a term of order zero may destroy the property of global hypoellipticity of operators of principal type, contrary to that happens with the usual (local) hypoellipticity.

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