Abstract
The Schwarz problem for \(J\) -analytic functions inan arbitrary ellipse is considered. The matrix \(J\) is assumed to betwo-dimensional with distinct eigenvalues lying above the real axis. An example of a nonconstantsolution of the homogeneous Schwarz problem in the form of a vector polynomial of degree three isgiven. A numerical parameter \(l\) of the matrix \(J\) , expressed via its eigenvectors, is introduced. Afterthat, one relation derived earlier by the present author is analyzed. Based on this analysis, amethod for computing the dimension and structure of the kernel of the Schwarz problem in anarbitrary ellipse is obtained. Sufficient conditions for the triviality of the kernel expressed via theellipse parameters, the eigenvalues of the matrix \(J\) , and the parameter \(l\) are obtained. Examples of one-dimensional andtrivial kernels are given.
