  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.
