Consider a smooth $n$-dimensional manifold $\mathcal{M}$ equipped with local coordinates $x$. In the tangent bundle $T\mathcal{M}$ we consider $n$ rank $1$ distributions $\mathcal{D}_i$ satisfying \begin{align*} \mathcal{D}_1(x_0)\oplus\ldots\oplus\mathcal{D}_n(x_0)=T\mathcal{M}. \end{align*}
Question: What are the conditions for the existence of local coordinates $x_i$ such that $dx_i \perp \mathcal{D}_j$ for all $i\neq j$, i.e. $\mathcal{D}_j=\cap_{i\neq j}\ker{dx_i}$.
Part of the answer: If $n=2$, then it is always possible without any extra assumptions. But I believe that this fact does not remain true in higher dimension.
Notations: Let $M^N$ be a smooth $N$-dimensional manifold equipped with local coordinates $\xi$. We consider $n$ involutives distributions $\{\mathcal{D}_1,\ldots,\mathcal{D}_n\}$ of constant rank, $\textrm{rk}(\mathcal{D}_i)=d_i$ satisfying $d:=\sum_{i=1}^{n}d_i\leq N$, and such that $$ \dim\left(\mathcal{D}_1(\xi_0)\oplus\ldots\oplus\mathcal{D}_n(\xi_0)\right)= d. $$ We denote $\mathcal{D}_{i_1,\ldots,i_k}$, for $k\leq n$, the distribution $\mathcal{D}_{i_1}\oplus\ldots\oplus\mathcal{D}_{i_k}$.
Proposition (Generalization of Frobenius Theorem): Under the above assumptions and notations. There exists coordinates $\xi=(x_1^1,\ldots,x_1^{d_1},\ldots,x_n^1,\ldots,x_n^{d_n})$ such that $\mathcal{D}_i=\textrm{span}\left\{\frac{\partial}{\partial x_i^1},\ldots,\frac{\partial}{\partial x_i^{d_i}}\right\}$ if and only if for all $i_1,i_2\in\{1,\ldots,n\}$ the distribution $\mathcal{D}_{i_1,i_2}$ is involutive.
The proof will use the following lemma.
Lemma: Under the above assumptions. If for all $i_1,i_2\in\{1,\ldots,n\}$ the distribution $\mathcal{D}_{i_1,i_2}$ is involutive then for any multi-index $i_1,\ldots,i_k$, with $k\leq n$, the distribution $\mathcal{D}_{i_1,\ldots,i_k}$ is involutive.
Proof: By a direct computation. For any $v,w\in\mathcal{D}_{i_1,\ldots,i_k}$ we have $v=v_{i_1}+\ldots+v_{i_k}$ with $v_{i_j}\in\mathcal{D}_{i_j}$ (similarly for $w$) and thus, $$ [v,w]=\left[\sum_{j=1}^kv_{i_j},\sum_{l=1}^kw_{i_l}\right]=\sum_{j=1}^k\sum_{l=1}^k \underbrace{[v_{i_j},w_{i_l}]}_{\in\mathcal{D}_{i_j,i_l}\subset\mathcal{D}_{i_1,\ldots,i_k}}. $$ $\square$
Proof of the Proposition: By the above lemma the distribution $\mathcal{D}_{1,\ldots,n}$ is involutive (and is of constant rank by assumption). Thus, we can reduce the proof in the submanifold $M^d:=M^N/\sim$ where $\sim$ is given by the involutive distribution $\mathcal{D}_{1,\ldots,n}$.
We will proceed by imbrication. By assumptions, the distributions $\mathcal{D_1}$ and $\mathcal{D}_{2,\ldots,n}$ are involutive and of constant rank $d_1$ and $d-d_1$ respectively. Using the answer in that post we construct coordinates such that $\mathcal{D}_1=\textrm{span}\left\{\frac{\partial}{\partial x^1_1},\ldots, \frac{\partial}{\partial x^{d_1}_1}\right\}$. Then we reduce the proof in the submanifold $M^{d-d_1}:=M^d/\sim$ where $\sim$ is given by the involutive distribution $\mathcal{D}_1$.
Finally we repeat the previous argument with the distributions $\mathcal{D}_2$ and $\mathcal{D}_{3,\ldots,n}$. And so on.
$\square$