Irreducible Fuchsian system with reducible monodromy representation

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.