Fuchsian systems with completely reducible monodromy

The solvability of the Riemann-Hilbert problem for representations χ = χ ₁ ⊕ χ ₂ having the form of a direct sum is considered. It is proved that any representation χ ₁ can be realized as a direct summand in the monodromy representation χ of a Fuchsian system. Other results are also obtained, which suggest a simple method for constructing counterexamples to the Riemann-Hilbert problem.