Let $U$ be an open subset of $\mathbb{R}^n$ and let $f\in C^1(U)$ be a continuous differentiable vector field $$f\colon U \to \mathbb{R}^n.$$ If there exists $x_0\in U$ such that $f(x_0)\neq 0$ then there exists $U_0\subset U$ an open neighborhood of $x_0$ such that there exists a diffeomorphism of class $C^1$ $$F\colon U_0 \to V$$ where $V$ is an open subset of $\mathbb{R}^n$ such that $$[DF(x)]f(x)=e_1$$ for all $x\in U_0$ where $[DF(x)]$ is the Jacobian of the diffeomorphism $F$ evaluated at $x$ and $e_1=(1,0,\ldots,0)$ is the first vector of the canonical basis of $\mathbb{R}^n$ . More generally if the vector field $f$ is of class $C^r$ then so is the diffeomorphism $F$ . [AVI]

References

AVI

Arnold, V.I.: Ordinary Differential Equations (translated by R.A. Silverman). The MIT Press, Cambridge, 1973.