  We study a real, massive Klein-Gordon field in the Poincar\'e fundamental
domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime, subject to a
particular choice of dynamical boundary conditions of generalized Wentzell
type, whereby the boundary data solves a non-homogeneous, boundary Klein-Gordon
equation, with the source term fixed by the normal derivative of the scalar
field at the boundary. This naturally defines a field in the conformal boundary
of the Poincar\'e fundamental domain of AdS. We completely solve the equations
for the bulk and boundary fields and investigate the existence of bound state
solutions, motivated by the analogous problem with Robin boundary conditions,
which are recovered as a limiting case. Finally, we argue that both Robin and
generalized Wentzell boundary conditions are distinguished in the sense that
they are invariant under the action of the isometry group of the AdS conformal
boundary, a condition which ensures in addition that the total flux of energy
across the boundary vanishes.
