Geodesic
Fuchsian Systems with Completely Reducible Monodromy
Matematičeskie zametki, Tome 85 (2009) no. 6, pp. 817-825

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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.
Keywords: Riemann–Hilbert problem, decomposable Fuchsian system, completely reducible monodromy, (semi)stable bundle with connection, holomorphic (meromorphic) function. 
I. V. Vyugin. Fuchsian Systems with Completely Reducible Monodromy. Matematičeskie zametki, Tome 85 (2009) no. 6, pp. 817-825. http://geodesic.mathdoc.fr/item/MZM_2009_85_6_a1/
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     title = {Fuchsian {Systems} with {Completely} {Reducible} {Monodromy}},
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