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Symmetry analysis can provide a suitable change of variables, i.e., ingeometric terms, a suitable diffeomorphism that simplifies the given directionfield, which can help significantly in solving or studying differentialequations. Roughly speaking this is the so-called rectification theorem. Thelocal version of this result is a well-known theorem in the field of ordinarydifferential equations. In this note we prove a global counterpart when theequation fulfils the Lipschitz condition. Then we use this result to determinethe global symmetry group of such an ordinary differential equation. It turnsout that, assuming the Lipschitz condition, the full symmetry group is a smoothwreath product of two diffeomorphism groups, and does not depend on the form ofthe equation, at all.

The global symmetry group of first order differential equations and the global rectification theorem