  In a previous paper, the authors together with L. Vrancken initiated the
study of $3$-dimensional CR submanifolds of the nearly K\" ahler homogeneous
$\mathbb S^3\times \mathbb S^3$. As is shown by Butruille this is one of only
four homogeneous $6$-dimensional nearly K\"ahler manifolds. Besides its almost
complex structure $J$ it also admits a canonical almost product structure $P$.
Along a $3$-dimensional CR submanifold the tangent space of $\mathbb
S^3\times\mathbb S^3$ can be naturally split as the orthogonal sum of three
$2$-dimensional vector bundles $\mathcal D_1$, $\mathcal D_2$ and $\mathcal
D_3$. We study the CR submanifolds in relation to the behavior of the almost
product structure on these vector bundles.
