  In this paper we give a characterization of real hypersurfaces in noncompact
complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, $m \geq 2$ with Reeb
vector field $\xi$ belonging to the maximal quaternionic subbundle $\mathcal
Q$. Then it becomes a tube over a totally real totally geodesic ${\mathbb
H}H^n$, $m=2n$, in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2
U_m)$, a horosphere whose center at the infinity is singular or another
exceptional case.
