  Let $\widetilde{J}$ be the canonical para-complex structure on
$\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine
hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field
(sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well
as $3$-dimensional $\widetilde{J}$-tangent affine hyperspheres with the
property that at least one null-direction of the second fundamental form
coincides with either $\mathcal{D}^+$ or $\mathcal{D}^-$. The main purpose of
this paper is to give a full local classification of the above mentioned
hypersurfaces. In particular, we prove that every nondegenerate centro-affine
hypersurface of dimension $3$ with a $\widetilde{J}$-tangent centro-affine
vector field which has two null-directions $\mathcal{D}^+$ and $\mathcal{D}^-$
must be both an affine hypersphere and a hyperquadric. Some examples of these
hypersurfaces are also given.
