Abstract
We study the global hypoellipticity property for non-singular planar complex vector fields. The results obtained are related to condition (P) of Nirenberg-Treves and the boundedness of certain subsets of the plane where the real and the imaginary parts of the vector field under consideration are linearly dependent. We also prove the existence of a globally defined first integral when the vector field is globally hypoelliptic and has only one orbit.
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