# differential equations – Finding a local normal form regarding distribution rank properties

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 ?