  Motivated by the problem of stability of Anti-de Sitter (AdS) spacetime, we
discuss nonlinear gravitational perturbations of maximally symmetric solutions
of vacuum Einstein equations in general and the case of AdS in particular. We
present the evidence that, similarly to the self-gravitating scalar field at
spherical symmetry, the negative cosmological constant allows for the existence
of globally regular, asymptotically AdS, time-periodic solutions of vacuum
Einstein equations whose frequencies bifurcate from linear eigenfrequencies of
AdS. Interestingly, our preliminary results indicate that the number of one
parameter families of time-periodic solutions bifurcating from a given
eigenfrequency equals the multiplicity of this eigenfrequency.
