On the dimension of the subspace of Liouvillian solutions of a Fuchsian system (Q1688324)

The author continues (see his work with \textit{I. V. Vyugin} [Russ. Math. Surv. 67, No. 3, 585--587 (2012; Zbl 1259.34090); translation from Usp. Mat. Nauk. 67, No. 3, 183--184 (2012); Arnold Math. J. 1, No. 4, 445--471 (2015; Zbl 1339.34091)]) his consideration of a Fuchsian system of linear differential equations on the Riemann sphere \(\bar{\mathbb{C}}\) of the form \[ \frac{dY}{dz}=\left( \sum\limits_{i=1}^{n}{\frac{{{B}_{i}}}{z-{{a}_{i}}}} \right)Y, \] where \({{a}_{1}},\ldots ,{{a}_{n}}\in \mathbb{C},{{B}_{1}},\ldots ,{{B}_{n}}\in {{M}_{p}}(\mathbb{C})\). If the Galois group of system (1) is reducible to a block-diagonal form then the matrix \(B=\sum\limits_{i=1}^{n}{\frac{{{B}_{i}}}{z-{{a}_{i}}}}\) can be also reduced to a block-diagonal form \({{B}^{*}}\) by a gage transformation \({{B}^{*}}=TB{{T}^{-1}}+{T}'{{T}^{-1}},T\in G{{L}_{F}}(n)\). It is shown in the paper that if the eigenvalues of the residue matrices satisfy certain conditions, we can select the transformation so that \(T\in G{{L}_{\mathbb{C}}}(n) \). In particular, this can be used when the system (1) admits Liouvillian solutions.