A detailed perturbative construction of globally regular, asymptotically anti–de Sitter (AdS) time-periodic solutions of Einstein’s equations with a negative cosmological constant (AdS geons) is presented. Starting with the most general superposition of the l=2 even parity (scalar) eigenmodes of AdS at linear order, it is shown that at the fifth order in perturbation theory one obtains five one-parameter geon families, two of which have a helical Killing vector, one with axial symmetry, and two others without continuous symmetries. The details and some subtle aspects of the perturbative expansions are also presented.