Let $M^n$ be a smooth manifold. Let $D$ be a (regular) integrable distribution on $M$.
The question is whether there always (i.e. for any $M$ and $D$) exists an integrable distribution $D'$ on $M$ which is pointwise complementary to $D$, i.e. at every point $p\in M$ we have $D_p \oplus D'_p = T_pM$.
Note: It is easy to prove that there exists a (not necessarily integrable) complementary distribution $D'$: one can take $D' := D^\bot$ with respect to a Riemannian metric on $M$. (There is a Riemannian metric on $M$ since by Whitney embedding theorem $M$ can be embedded in $\mathbb{R}^{2n}$ and hence can be equipped with the metric induced from the standard scalar product.) However, it is not clear why or whether $D'$ is integrable.