  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 $M$.
