  In this paper we study ACM vector bundles $\E$ of rank $k \geq 3$ on
hypersurfaces $X_r \subset\Pj^4$ of degree $r \geq 1$. We consider here mainly
the case of degree $r = 4$, which is the first unknown case in literature.
Under some natural conditions for the bundle $\E$ we derive a list of possible
Chern classes $(c_1,c_2,c_3)$ which may arise in the cases of rank $k=3$ and
$k=4$, when $r=4$. For some cases among these we give the corresponding
examples, the existence of all the other cases remaining under question.
