Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians

We classify the real hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians _SU2,m/S(_U2⋅_Um)${\mathit{SU}}_{2,m}/S(U_2\cdot U_m)$, m?2$m?2$. Each can be described as a tube over a totally geodesic _SU2,m−1/S(_U2⋅_Um−1)${\mathit{SU}}_{2,m-1}/S(U_2\cdot U_{m-1})$ in _SU2,m/S(_U2⋅_Um)${\mathit{SU}}_{2,m}/S(U_2\cdot U_m)$ or a horosphere whose center at infinity is singular.