Regular systems whose monodromies differ from monodromies of Fuchsian systems (Q1917798)

As is known, not every representation (1) \(\chi: \pi_1(\mathbb{C}\mathbb{P}^1 \setminus \{a_1,\dots,a_n\}, z_0)\to\text{GL} (p;\mathbb{C})\) can be realized as the monodromy representation of a Fuchsian system \[ df= \Biggl( \sum^n_{i=1} B_i{{dz}\over {z-a_i}}\Biggr)f, \tag{2} \] where \(B_i\) are \(p\times p\) constant matrices. However, a similar problem for the class of systems with regular singular points always has a positive solution. In other words, there are systems (3) \(df=\omega f\) with regular points \(a_1,\dots,a_n\) that cannot be reduced to the Fuchsian system (2) by any change (4) \(\Gamma(z) f=g\) of the unknown vector function \(f\) by means of the matrix \(\Gamma(z)\) meromorphic on \(\mathbb{C}\mathbb{P}^1\) which is holomorphically invertible outside of the points \(a_1,\dots,a_n\). The following theorem is the main result of the paper. Theorem 1. Any system (3) with regular singular points is a subsystem (a quotient system) of some system with the same singular points meromorphically equivalent to the Fuchsian system (2). Theorem 1 is equivalent to the following statement. Theorem 2. Any representation (1) is a subrepresentation (a quotient representation) of some representation for which the Riemann-Hilbert problem has a positive solution.