Let $M$ be a n dimension manifold, $X\in\mathscr{X}^1(M)$ and $f$ a first integral for the flux generated by $X$. Assuming $f$ is of $C^1$ class and $Df(p)\neq 0$, show that exists a neigborhood of $p$ such that $X$ is topologically conjugate of a flux of the form $Y=(Y_1,Y_2,...,Y_{n-1},0)$ in $\mathbb{R}^n$.
Definition: A first integral with real values $f:U\subset M\to\mathbb{R}$ is said to be a first integral for a flux $(\phi_t)_t$ if:
i) f is flux invariant
ii) f is not constant in open subsets of $U$.
Attempted proof:
Consider $F$ the graphic:
$F:M\to \mathbb{R}^n\\(x_1,x_2,...,x_{n-1},\alpha)\to(x_1,x_2,...,x_{n-1},f(x_1,..x_{n-1}))$
I could apply the implicit function theorem to assure me a open neighbourhood $V\subset M$ for $p_1,...,p_2$ if I can assure that $\det Df(p)\neq 0$(so that $Df(p)$ is a linear isomorphism). However the question only concedes me the fact $Df(p)\neq 0$ not the determinant.
Question:
1) Is there any way to conlude $\det Df(p)\neq 0$? Or should I search for another way to prove the assertion?
2) Can I subtract a constant $\alpha$ to get $(y_1,...y_{n-1},0)$?
Thanks in advance!