On the solutions of the Schwartz homogeneous problem in the form of vector polynomials of the second degree

A Schwartz homogeneous problem for vector functions being analytic as per Douglis in the final domain on plane has been considered. The function data are the solutions to a homogeneous elliptic partial-derivatives system of the first order, which depends on the matrix with complex coefficients. It is assumed that the determinant of the complex part of this matrix is different from zero. In the beginning of the article the solution to the problem under consideration is given for a bivariate case in the form of a vector polynomial of the second degree. Further transformations are made to find the general method of building such solutions. It is demonstrated that the real part of the function being analytic as per Douglis will be a solution for a certain homogeneous partial-derivatives system of the second order. Meanwhile, if we know the solution to the Dirichlet problem for this system, we can build the solution to the Schwartz problem, corresponding to the initial matrix. We are searching for the required solution to the Dirichlet problem in the form of a vector polynomial of the second degree with linearly-dependent components. After this function's substitution into the obtained system of partial-derivatives equations, we get a real homogeneous algebraic system. Such system will have nonzero solutions only in case its determinant is equal to zero. By setting the relevant determinant to zero, we get a two-variable algebraic equation. Further on, we prove a fundamental theorem that the existence of a real arbitrary nonzero solution to this algebraic equation is a necessary and sufficient condition of the existence the solution to the Schwartz homogeneous problem in the form of a vector polynomial of the second degree, corresponding to the initial matrix. In the end of the article we give an example of using the fundamental theorem to build two solutions to the studied problem, which correspond to the set matrix.