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We consider a flow of the plane which has no fixed points. We present a method for finding a countable family of maximal parallelizable regions of the flow which cover the plane. Moreover, we describe the relations between parallelizing homeomorphisms defined on the maximal parallelizable regions using solutions of appropriate functional equations.doi:10.5486/pmd.2009.4423 fatcat:jh4d54g4mrfsxlwnwhhbv6woli