On linearly independent solutions of the homogeneous Schwartz problem

We study the homogeneous Schwarz problem for Douglis analytic functions. We consider two-dimensional matrices $J$ with a multiple eigenvalue and the eigenvector, which is not proportional to a real vector. We obtain a sufficient condition for the matrix $J$ under which there exist two linearly independent solutions of the problem defined in a certain domain $D$. We present an example.