  The meson $X(3872)$ and several related states appear to be, at least in
part, hadronic molecules in which a heavy flavored meson (such as $D^0$) is
bound to another heavy meson (such as $\bar D^{*0}$). Although not the only
effect contributing to the binding, pion exchange seems to play a crucial role
in generating the longest-range force between constituents. Mesons without $u$
and $d$ light quarks (such as $D_s$) cannot exchange pions, but under suitable
circumstances can bind as a result of $\eta$ exchange. Channels in which this
mechanism is possible are identified, and suggestions are made for searches for
the corresponding molecular states, including a manifestly exotic baryonic
$\Lambda_c \bar D_s^*$ resonance decaying into $J/\psi~\Lambda$.
