These isometries of AdS are very powerful.

Let us recall the situation in flat space. If we have a massive geodesic in flat space then we can always boost to a frame where the particle is at rest. In AdS it is the same: if we consider the oscillating trajectory of a massive particle then we can "boost" to a frame where the particle is at rest.

Thus, the moving particle does not know that is moving and, despite appearances, there is no "center" in AdS. The Hamiltonian is part of the symmetry group (as in the Poincaré group) and there are several choices of Hamiltonian. Once we choose a Hamiltonian ... then we have chosen a "center" and a notion of the lowest energy state, in which a particle sits at this "center.