  We consider the conformal wave equation on the Einstein cylinder with a
defocusing cubic non-linearity. Motivated by a method developed by
Rostworowski-Maliborski on the existence of time periodic solutions to the
spherically symmetric Einstein-Klein-Gordon system, we study perturbations
around the zero solution as a formal series expansion and assume that the
perturbations bifurcate from $1-$mode. In the center of this work stands a
rigorous proof on how one can choose the initial data to cancel out all secular
terms in the resonant system. Interestingly, our analysis reveals that the only
possible choice for the existence of time periodic solutions is when the error
terms in the expansion are all proportional to the dominant $1-$mode. Finally,
we use techniques from ordinary differential equations and establish the
existence of time periodic solutions for initial data proportional to the first
mode of the linearized operator.
