| Summary: | The Schwarz problem for <i>J</i>-analytic functions in an ellipse is considered. In this case, the matrix <i>J</i> is assumed to be two-dimensional with different eigenvalues located above the real axis. The Schwarz problem is reduced to an equivalent boundary value problem for the scalar functional equation depending on the real parameter <i>l</i>. This parameter is determined by the Jordan basis of the matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>.</mo></mrow></semantics></math></inline-formula> An analysis of the functional equation was performed. It is shown that for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, the solution of the Schwarz problem with matrix <i>J</i> exists uniquely in the Hölder classes in an arbitrary ellipse.
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