  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.
