  We consider the $(1 + 3)$-dimensional Einstein equations with negative
cosmological constant coupled to a spherically-symmetric, massless scalar field
and study perturbations around the Anti-de Sitter spacetime. We derive the
resonant systems, pick out vanishing secular terms and discuss issues related
to small divisors. Most importantly, we rigorously establish (sharp, in most of
the cases) asymptotic behaviour for all the interaction coefficients. The
latter is based on uniform estimates for the eigenfunctions associated to the
linearized operator as well as on some oscillatory integrals.
