In the ODE given by:
$x'=X(x)$ , where $X$ is my vectorial field $\in C^1$ in an open subset of $\mathbb R^n$ ,
If $X$ have $f$ a first integral and $df(p)\neq0$ then there exists a neighborhood $V_{p}$ of $p$ such that $X\restriction V_{p}$ is differenciably conjugated to a system:
$Y=(Y_{1},Y_{2},...,Y_{n-1},0)$
So far i tried showing such a conjugation, i know that for every $c$ constant of $f$ the surface (here i don`t know how to call it, manifold maybe) $f^{-1}(c)$ is regular, so there is a map with rank $n-1$, this was my first thought, but i can't build such a conjugation
Also i thought of building such a conjugation withthe gradient of $f$, also couldn't succed
Basically what the question states is that if a system have a first integral then it can be reduced by a simpler system.
Any thoughts?