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. 
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     author = {I. V. Vyugin},
     title = {Fuchsian {Systems} with {Completely} {Reducible} {Monodromy}},
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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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