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In this paper we investigate compatible overdetermined systems of PDEs on theplane with one common characteristic. Lie's theorem states that its integrationis equivalent to a system of ODEs, and we relate this to the geometry of rank 2distributions. We find a criterion for integration in quadratures and in closedform, as in the method of Darboux, and discuss nonlinear Laplacetransformations and symmetric PDE models.

Lie theorem via rank 2 distributions (integration of PDE of class \omega=1)