On the study of the Neumann problem for elliptic system of two sixth order equations on the plane

As it is known, on the basis of the methods of the theory of singular integral equations, fine results were obtained in the theory of partial differential equations. In this paper, we study the question of solvability of the Neumann problem for an elliptic system of two six order equations with two independent variables in a bounded domain. During the study of this problem, the method developed by Boyarsky is used. The essence of this method is to construct a matrix function on base of the main part of the given system and split polynomials into homotopy classes. Using this approach, the ellipticity of the system under consideration is proved. It is also shown that, in accordance with homotopy classes, an elliptic system of two sixth order equations with two independent variables can be equivalently reduced to a singular integral equation over a bounded domain. Using the method of passing to an equivalent singular integral equation over a bounded domain, effective Noetherian conditions are found, and a formula for calculating the index of the problem is obtained.