  We study germs of holomorphic distributions with "separated variables'. In
codimension one, a well know example of this kind of distribution is given by
the canonical contact structure on $\mathbb{P}^{2m+1}$ . Another example is the
Darboux distribution, which gives the normal local form of any contact
structure. Given a germ $D$ of holomorphic distribution with separated
variables in $(\mathbb{C}^n,0)$, we show that there exists , for some $\kappa
\in \mathbb{Z}_{\geq 0}$ related to the Taylor coefficients of $D$, a
holomorphic submersion $H_{D}: (\mathbb{C}^n,0) \rightarrow
(\mathbb{C}^{\kappa},0)$ such that $D$ is completely non-integrable on each
level of $H_{D}$. Furthermore, we show that there exists a holomorphic vector
field $Z$ tangent to $D$, such that each level of $H_{D}$ contains a leaf of
$Z$ that is somewhere dense in the level. In particular, the field of
meromorphic first integrals of $Z$ and that of $D$ are the same.
