A variational formulation of the synthesis problem for plane antenna systems according to the prescribed power radiation pattern (RP) is considered. The functional representing the mean-square deviation of the prescribed and synthesized power RPs and containing the additional term with squared norm of the current or field in the antenna aperture is considered as criterion of optimization. Freedom to choose the phase RP is used to improve the proximity of the prescribed and synthesized RPs. In such formulation, the classes of nonlinear problems, for which the non-uniqueness of solutions, their branching and bifurcation are characteristic, arise. The properties of solutions depend on the electric size of antenna system and prescribed power RP. From a practical point of view, the existence of different solutions creating the same or similar RPs, gives the opportunity to choose the solution that has a simpler implementation. The specificity of statement of the synthesis problem is considered for several types of antenna systems.