Splitting aspects of holomorphic distributions with locally free tangent sheaf

In this work, we mainly deal with a two-dimensional singular holomorphic distribution D defined on M, in the two situations M=Pⁿ or M=(Cⁿ,0), tangent to a one-dimensional foliation G on M, and whose tangent sheaf TD is locally free. We provide sufficient conditions on G so that there is another one-dimensional foliation H on M tangent to D, such that their respective tangent sheaves satisfy the splitting relation TD=TG ⊕ TH. As an application, we show that if F is a codimension one holomorphic foliation on P³ with locally free tangent sheaf and tangent to a nontrivial holomorphic vector field on P³, then TF splits. Some results on division of holomorphic differential forms by tangent vector fields are also obtained.