On the binding of the $BD\bar{D}$ and $BDD$ systems

We study theoretically the $BD\bar{D}$ and $BDD$ systems to see if they allow for possible bound or resonant states. The three-body interaction is evaluated implementing the Fixed Center Approximation to the Faddeev equations which considers the interaction of a $D$ or $\bar{D}$ particle with the components of a $BD$ cluster, previously proved to form a bound state. We find an $I(J^P)=1/2(0^-)$ bound state for the $BD\bar{D}$ system at an energy around $8925-8985$ MeV within uncertainties, which would correspond to a bottom--hidden-charm meson. In contrast, the $BDD$ system, which would be bottom--double-charm and hence manifestly exotic, we have found hints of a bound state in the energy region $8935-8985$ MeV, but the results are not stable under the uncertainties of the model, and we cannot assure, neither rule out, the possibility of a $BDD$ three-body state.