Canonical form of a regular vector field

Also called the "flow box coordinates theorem".

See @lee2013smooth.

It is generalized to the canonical form of commuting vector fields.

It can be expressed in term of first integrals, also:

Given the open set $U\subseteq {\mathbb{R}}^n$, $X\in \mathfrak{X}(U)$ and $x_0\in U$ such that $X(x_0)\ne 0$, we can find an open subset $x_0\in V\subseteq U$ such that the system

admits $n-1$ first integrals which are functionally independent and such that any other first integral depends functionally on them. (Proof [Arnold 1991]).

This $n-1$ first integrals $W_i$ define a curve in $V\subseteq {\mathbb{R}}^n$ that is precisely a solution curve $\gamma (t)$. Moreover, the $n-1$ first integrals together with the first coordinate function let us define a coordinate change

so that the vector field $X$ is rectificated. Because of this this result is also called rectification lemma for vector fields.