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. 
@article{MZM_2006_80_4_a2,
     author = {I. V. Vyugin},
     title = {Irreducible {Fuchsian} system with reducible monodromy representation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {501--508},
     publisher = {mathdoc},
     volume = {80},
     number = {4},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_4_a2/}
}
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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/