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In this paper we give a characterization of real hypersurfaces in noncompactcomplex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, $m \geq 2$ with Reebvector field $\xi$ belonging to the maximal quaternionic subbundle $\mathcalQ$. Then it becomes a tube over a totally real totally geodesic ${\mathbbH}H^n$, $m=2n$, in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2U_m)$, a horosphere whose center at the infinity is singular or anotherexceptional case.

Real hypersurfaces in complex hyperbolic two-plane Grassmannians related to the Reeb vector field