Fuchsian Systems with Completely Reducible Monodromy

The solvability of the Riemann–Hilbert problem for representations $\chi=\chi_1\oplus\chi_2$ having the form of a direct sum is considered. It is proved that any representation $\chi_1$ can be realized as a direct summand in the monodromy representation $\chi$ of a Fuchsian system. Other results are also obtained, which suggest a simple method for constructing counterexamples to the Riemann–Hilbert problem.