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

On $3$-dimensional $\widetilde{J}$-tangent centro-affine hypersurfaces and $\widetilde{J}$-tangent affine hyperspheres with some null-directions