Geodesic
Irreducible Fuchsian system with reducible monodromy representation
Matematičeskie zametki, Tome 80 (2006) no. 4, pp. 501-508.

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We present an example of the reducible representation $\chi=\chi_1\oplus\chi_2$, which, on the one hand, is the monodromy representation of a Fuchsian system. On the other hand, the representation $\chi_2$ is a counterexample to the Riemann–Hilbert problem. Using a meromorphic gauge transformation, one cannot reduce this system to the direct sum of Fuchsian systems corresponding to the subrepresentations. 
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I. V. Vyugin. Irreducible Fuchsian system with reducible monodromy representation. Matematičeskie zametki, Tome 80 (2006) no. 4, pp. 501-508. http://geodesic.mathdoc.fr/item/MZM_2006_80_4_a2/

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