Conservation laws and solutions of the first boundary value problem for equations of two-dimensional elasticity theory

If a system of differential equations admits a group of continuous transformations, then the system can be represented as a set of two systems of differential equations, it is known. As a rule, these systems have a smaller order than the original system. The first system is automorphic, characterized by the fact that all its solutions are obtained from a single solution using transformations of this group. The second system is permissive, its solutions, under the action of the group, pass into themselves. The resolving system carries basic information about the source system. Automorphic and resolving systems, two-dimensional stationary elasticity equations are studied in this work. They are systems of first-order differential equations. Infinite series of conservation laws for a resolving system of equations and an automorphic system are constructed for the first time in this work. Since the two-dimensional system of elasticity equations is linear, there are infinitely many such laws. In this paper, an infinite series of linear conservation laws with respect to the first derivatives is constructed. It was these laws that made it possible to solve the first boundary value problem for the equations of elasticity theory in the two-dimensional case. These solutions are constructed in the form of quadratures, these quadratures are calculated along the contour of the studied area.