  We consider the resonant system of amplitude equations for the conformally
invariant cubic wave equation on the three-sphere. Using the local bifurcation
theory, we characterize all stationary states that bifurcate from the first two
eigenmodes. Thanks to the variational formulation of the resonant system and
energy conservation, we also determine variational characterization and
stability of the bifurcating states. For the lowest eigenmode, we obtain two
orbitally stable families of the bifurcating stationary states: one is a
constrained maximizer of energy and the other one is a constrained minimizer of
the energy, where the constraints are due to other conserved quantities of the
resonant system. For the second eigenmode, we obtain two constrained minimizers
of the energy, which are also orbitally stable in the time evolution. All other
bifurcating states are saddle points of energy under these constraints and
their stability in the time evolution is unknown.
