represented in analytic form in terms of series with respect to certain group of symmetries which are obtained by using functional equation method. Thus, at the study of properties of circular composites with constant conductivity of each component one can use an analytic formulas for the solution as well as analytic formulas for effective conductivity [1]. When the multiply connected domain tends to the limiting punctured domain the elliptic state equation becomes degenerating at a finite

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... r of points. Thus we have to perform an asymptotic analysis of the solution to the corresponding boundary value problems. It is difficult to refer to all essential contribution to the asymptotic study of the solution to the boundary value problems for elliptic equations in singularly perturbed domain. Recently a new approach has been proposed (see, e.g., [2]) based on the methods of nonlinear functional analysis. In our work we use also a special type of asymptotic analysis (see [3]) performed for the case boundary value problems for elliptic equations degenerating at a line (in three-dimensional case) or at a point (in two-dimensional case). ; a i, λ p ij , α p-parameters of the system; q-the number of additional equation connecting arc flow for proceeding arc from through the set split ratio coefficients; x = (x ij , (i, j) ∈ U ; x i , i ∈ I *)-vector of unknowns. Calculation of a rank of the matrix of the system (3)-(4), building the algorithms for finding the solutions of the systems of the type (3)-(4) and characteristics of optimal solutions are investigated in [2]-[4].