
Journal of Lie Theory 31 (2021), No. 1, 237247 Copyright Heldermann Verlag 2021 The Symmetry Group of First Order Differential Equations and the Global Rectification Theorem Eszter Gselmann Institute of Mathematics, University of Debrecen, 4002 Debrecen, Hungary gselmann@science.unideb.hu Gábor Horváth Institute of Mathematics, University of Debrecen, 4002 Debrecen, Hungary ghorvath@science.unideb.hu Symmetry analysis can provide a suitable change of variables, i.e., in geometric terms, a suitable diffeomorphism that simplifies the given direction field, which can help significantly in solving or studying differential equations. Roughly speaking this is the socalled rectification theorem. The local version of this result is a wellknown theorem in the field of ordinary differential equations. In this note we prove a global counterpart when the equation fulfils the Lipschitz condition. Then we use this result to determine the global symmetry group of such an ordinary differential equation. It turns out that, assuming the Lipschitz condition, the full symmetry group is a smooth wreath product of two diffeomorphism groups, and does not depend on the form of the equation, at all. Keywords: Global rectification, symmetry, global existence and uniqueness theorem, symmetries of differential equations, wreath product, diffeomorphism group. MSC: 34A12, 34C40, 22E99, 20E22. [ Fulltextpdf (132 KB)] for subscribers only. 