Linear differential systems with small coefficients: various types of solvability and their verification (Q2330830)

The paper generalizes some results (see [\textit{R. R. Gontsov} and \textit{I. V. Vyugin}, Arnold Math. J. 1, No. 4, 445--471 (2015; Zbl 1339.34091)]) on solvability in quadratures for systems of linear differential equations of the form \(\frac{dY}{dz}=B(z)Y,\quad B\in\mathbf{M}(n,\mathbb{C}\mathrm{(z))}\) with \textit{sufficiently small} coefficients. Such systems have some ``rigidity'' with respect to gage transformations. Namely, if \(\exists(S\in\mathbf{GL}(n,\overline{\mathbb{C}\mathrm{(z)}}))\) such that \(SBS^{-1}+\delta SS^{-1}\in\mathbf{T}(n,\overline{\mathbb{C}\mathrm{(z)}}))\) then \(\exists(K\in\mathbf{GL}(n,\mathbb{C}\mathrm{)})\) such that \(KBK^{-1}\in\mathbf{T}(n,\mathbb{C}(z))\). The authors show that the matrix-coefficients of the system at singular points generate a solvable Lie algebra if this system is solvable in quadratures. Then they demonstrate by examples how using this circumstance and the procedures from Maple, one can solve the problem of solvability in quadratures of the systems in question.