Addendum to the paper "Hypersurfaces with Isometric Reeb Flow in Complex   hyperbolic Two-Plane Grassmannians"
  We classify all of real hypersurfaces $M$ with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m \geq 2$. Then it becomes a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a horosphere whose center at infinity is singular and of type $JN \in {\mathfrak J}N$ for a unit normal vector field $N$ of $M$. 
Hyunjin Lee, Mi Jung Kim and Young Jin Suh
math.DG