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The nonlinear equation of a synthesis problem for the waveguide array using a standard technique of the calculus of variations is obtained. Utilization of the Bubnov-Galerkin method for the solution of a direct problem (analysis problem) allows for certain restrictions to obtain a real kernel of this equation. The existence of a real kernel enables to carry out analytical investigation of the non-uniqueness and branching of solutions of the obtained equation. Numerical results of research of the eigen values of the corresponding-linear equation are given.