Irreducible Fuchsian system with reducible monodromy representation (Q881132)

Problem (Bolibrukh): Consider the system \[ {dy\over dz}= \Biggl(\sum^n_{i=1} {B_i\over z- a_i}\Biggr) y,\quad \sum^n_{i=1} B_i= 0 \] on the Riemann sphere with singular points \(a_1,\dots, a_n\), whose monodromy representation \(\chi: \pi_1(\overline{\mathbb{C}}\setminus\{a_1,\dots, a_n\})\to \text{GL}(p, \mathbb{C})\) has the form \(\chi= \chi_1\oplus \chi_2\). Is it possible, using the transformation \(f= \Gamma y\), where \(\Gamma\) is a matrix function meromorphic at the points \(a_1,\dots, a_n\) and homomorphically invertible in \(\overline{\mathbb{C}}\setminus\{a_1,\dots, a_n\}\), to reduce any such system to the block-diagonal form \[ {df\over dz}= \Biggl(\sum^n_{i=1} {B_i'\over z- a_i}\Biggr) f, \] where \(B_i'= B^1_i\oplus B^2_i\) and \(B^1_i\) and \(B^2_i\) are the blocks corresponding to the subrepresentations \(\chi_1\) and \(\chi_2\)? This paper negatively answers this problem by giving an example: \(n= 5\), \(\dim\chi_1= 2\), \(\dim\chi_2= 4\). The Riemann-Hilbert problem for \(\chi_1\oplus\chi_2\) is solvable, while not for \(\chi_2\).