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

Abstract

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.