  In this work, we mainly deal with a two-dimensional singular holomorphic
distribution $\mathcal{D}$ defined on $M$, in the two situations
$M=\mathbb{P}^n$ or $M=(\mathbb{C}^n,0)$, tangent to a one-dimensional
foliation $\mathcal{G}$ on $M$, and whose tangent sheaf $T_{\mathcal{D}}$ is
locally free. We provide sufficient conditions on $\mathcal{G}$ so that there
is another one-dimensional foliation $\mathcal{H}$ on $M$ tangent to
$\mathcal{D}$, such that their respective tangent sheaves satisfy the splitting
relation $T_{\mathcal{D}}=T_{\mathcal{G}} \oplus T_{\mathcal{H}}$. As an
application, we show that if $\mathcal{F}$ is a codimension one holomorphic
foliation on $\mathbb{P}^3$ with locally free tangent sheaf and tangent to a
nontrivial holomorphic vector field on $\mathbb{P}^3$, then $T_{\mathcal{F}}$
splits. Some division results for vector fields and differential forms are also
obtained.
