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 $nleq 2m+1$. I know that there exists an involutive sub-distribution $mathscr{L}subsetmathscr{D}$ with rank $m$. I also know that the growth vector is $(m+1,n)$. (So I guess $mathscr{D}+(mathscr{D},mathscr{D})=Tmathbb{R}^n$, 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.

Reference: Article from Williams Pasillas and Witold Respondek (https://arxiv.org/abs/math/0004124)