  This paper is devoted to the study of holomorphic distributions of dimension
and codimension one on smooth weighted projective complete intersection Fano
manifolds threedimensional, with Picard number equal to one. We study the
relations between algebro-geometric properties of the singular set of singular
holomorphic distributions and their associated sheaves. We characterize either
distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen
Macaulay (aCM) on smooth weighted projective complete intersection Fano
manifolds threefold. We also prove that a codimension one locally free
distribution with trivial canonical bundle on any Fano threefold, with Picard
number equal to one, has a tangent sheaf which either splits or it is stable.
