An everyday task in Singularities is to build a local coordinate change (e.g. analytic) with prescribed properties. The classical approach is to cook up a special vector field, and to take its time-one-flow. (A highly non-algebraic/not o-minimal operation!)

In zero characteristic this 'time-one-flow' is defined as exp[vector field]. In many cases the vector field can be chosen 'in a nice class', e.g. polynomials/algebraic power series/holonomic functions/higher Picard-Vessiot extensions/Denjoy-Carleman/ $C^\infty$.

• 'Very rarely' in cases close to algebraic. (The criteria are of Liouville type.)
• 'Very often' for normed rings.
• Never in the $C^\infty$ case.