On one method for constructing solutions to the homogeneous schwarz problem

The paper considers the homogeneous Schwarz problem for Douglis analytic or $J$-analytic functions. The 2-2-matrix $J$ has eigenvalues $\lambda$,$\mu$, lying above the real axis. The eigenvalues can be either distinct or multiples. In the second section of the paper there are given the problem statement and definitions of $J$-analytic and $\lambda$-holomorphic functions. At the beginning of the third section Lemma 3.1 is proved, establishing some relation between real and holomorphic functions. Then is constructed a special operator basis $J$. For matrices with multiple eigenvalue this basis coincides with the Jordan basis of the matrix $J$. Then with the help of this basis and Lemma 3.1 is constructed $J$-analytic function $\phi(z)$ in the form of a quadratic vector polynomial of some special form. If the eigenvalues $\lambda$,$\mu$ of the matrix $J$ are fixed, then the function $\phi(z)$ depends on the elements of the first column of the matrix $J$ as $parameters. These parameters are chosen so that the real part of the function $\phi(z)$ has the form $(P; 0$), where $P = P(x, y)$ is a positively defined quadratic form. All $J$-analytic functions are defined with the accuracy of the additive vector constant Therefore, the function $\phi(z) - (1, 0)$ will be the required solution to the homogeneous Schwarz problem in the ellipse &\Gamma; : P(x, y) = 1$. Then the matrix $J$ is reconstructed by the known elements of the first column and the eigenvalues $\lambda$,$\mu$. The obtained result is formalized in the form of Theorem 3.1. At the end of the paper there are given six examples constructed according to the algorithm described above.