  Nirenberg's famous complex Frobenius theorem gives necessary and sufficient
conditions on a locally integrable structure for when the manifold is locally
diffeomorphic to $\mathbb R^r\times\mathbb C^m\times \mathbb R^{N-r-2m}$
through a coordinate chart $F$ in such a way that the structure is locally
spanned by $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partial
t^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$,
where we have given $\mathbb R^r\times\mathbb C^m \times\mathbb R^{N-r-2m}$
coordinates $(t,z,s)$. In this paper, we give the optimal H\"older-Zygmund
regularity for the coordinate charts which achieve this realization. Namely, if
the structure has H\"older-Zygmund regularity of order $\alpha>1$, then the
coordinate chart $F$ that maps to $\mathbb R^r\times\mathbb C^m \times\mathbb
R^{N-r-2m}$ may be taken to have H\"older-Zygmund regularity of order $\alpha$,
and this is sharp. Furthermore, we can choose this $F$ in such a way that the
vector fields $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partial
t^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$ on
the original manifold have H\"older-Zygmund regularity of order
$\alpha-\varepsilon$ for every $\varepsilon>0$, and we give an example to show
that the regularity for $F^*\frac\partial{\partial z}$ is optimal.
