I am working in geometry control field, fall last week on this exercice and I can't figure it out. I have a distribution $\mathscr{D}$ with $rank(\mathscr{D})=m+1$ in $\mathbb{R}^n$ with $n\leq 2m+1$. I know that there exists an involutive sub-distribution $\mathscr{L}\subset\mathscr{D}$ with rank $m$. I also know that the growth vector is $(m+1,n)$. (So I guess $\mathscr{D}+[\mathscr{D},\mathscr{D}]=T\mathbb{R}^n$, am I right?).
What I need is to find a local normal form $\varphi$ of $\mathscr{D}$ so that $\varphi_*\mathscr{D}=span\{f'_1,\dots,f'_n\}$ assuming $\mathscr{D}=span\{f_1,\dots,f_n\}$.
So far, I guess I can use the fact that $\mathscr{F}$ is involutive to write it $\mathscr{F}=span\{\frac{\partial}{\partial x^1},\dots,\frac{\partial}{\partial x^m}\}$ and then add a vector to build $\mathscr{D}$ but I don't know how. I guess Fröbenius can provide some help but can't find how. Someone has an idea ?
Thanks in advance.
Edit:
Reference: Article from Williams Pasillas and Witold Respondek (https://arxiv.org/abs/math/0004124)