I will start with a bit of general review of the AdS stability program exploring the elaborate nonlinear dynamics of small perturbations in Anti-de Sitter spacetime. The main mathematical tool for these investigations is nonlinear perturbation theory based on application of the time-averaging method. I will then focus on a specific simple realization of this setup (conformally coupled self-interacting probe scalar in AdS), for which the resulting effective time-averaged dynamics displays remarkable features connecting to the "cubic Szegő equation" (actively studied in the current math literature) and suggestive of integrability. I'll also mention connections to weakly nonlinear dynamics of trapped Bose-Einstein condensates.