Schwarz problem for $J$-analytic functions in an ellipse

The Schwarz problem for functions analytic in the sense of Douglis in an ellipse is considered. Necessary and sufficient conditions on the $l\times l$ matrix $J$ and the ellipse $\Gamma$ are obtained under which the Schwarz problem has a unique solution in Hölder classes. In the case of $l$ = 2 and matrices with distinct eigenvalues, the Schwarz problem is reduced to a scalar functional equation. Sufficient conditions on a Jordan basis of $J$ are obtained under which the Schwarz problem is solvable in an arbitrary ellipse. Matrices $J$ with eigenvalues lying above and below the real line are considered.