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Journal of Lie Theory 31 (2021), No. 1, 237--247
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

Gábor Horváth
Institute of Mathematics, University of Debrecen, 4002 Debrecen, Hungary

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 so-called rectification theorem. The local version of this result is a well-known 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.

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