A class of four dimensional CR submanifolds of the nearly Kähler six sphere

A submanifold $M$ of the nearly Kähler sphere $S^6(1)$ is called a $CR$-submanifold if there exists a $C^\infty$-differential almost complex distribution $U\: x \mapsto U_x \subset T_xM$, i.e., $JU = U$ on $M$, such that its orthogonal complement $U^\perp$ in $TM$ is totally real distribution, i.e., $JU^\perp \subset T^\perp M$, where $T^\perp M$ is the normal bundle over $M$ in $S^6(1)$. Since the four dimensional $CR$-submanifolds of $S^6(1)$ can not be totally geodesic, we investigate four dimensional $CR$-submanifolds that admit the distribution $D(p) = \{X \in T_pM\mid h(X, Y ) = 0,\ \forall Y \in T_pM\}$, of the maximal possible dimension which is two and classify them using sphere curves and vector fields along those curves.