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The problem of construction of numerical bilateral approximation for determination of branching points of one nonlinear integral operator, arising in the theory of antennas synthesis according to the given amplitude directivity pattern, is considered. The basic difficulty consists in that the kernel of integral operator nonlinearly depends on the parameter, which play role of the spectral one. Thus the problem is reduced to a nonlinear eigenvalue problem with application the technique of the alternating approximations of eigenvalues. The technique is based on a generalization of the known Rayleigh ratio for iinear problem onto nonlinear (initial and some auxiliary) eigenvalue problems. These generalized Rayleigh ratioes are used for constructing an iterative process of alternating eigenvalue approximations.