  In this paper we study $\widetilde{J}$-tangent affine hyperspheres, where
$\widetilde{J}$ is the canonical para-complex structure on $\mathbb{R}^{2n+2}$.
The main purpose of this paper is to give a classification of
$\widetilde{J}$-tangent affine hyperspheres of an arbitrary dimension with an
involutive distribution $\mathcal{D}$. In particular, we classify all such
hyperspheres in the $3$-dimensional case. We also show that there is a direct
relation between $\widetilde{J}$-tangent affine hyperspheres and Calabi
products. As an application we obtain certain classification results. In
particular, we show that, with one exception, all odd dimensional proper flat
affine hyperspheres are, after a suitable affine transformation,
$\widetilde{J}$-tangent. Some examples of $\widetilde{J}$-tangent affine
hyperspheres are also given.
