On the Poincaré problem solution for a system of two functions (Q6140437)

The paper presents the results of calculating the matrices of the monodromy group for the Fuchsian system of differential equations of the form \(\dfrac{dX} {dz}=\sum_{k=1}^{n}\dfrac{U_{k}}{z-a_{k}}\) for \(n=3,4.\) For example, the next theorem was obtained for the system (1) \(\dfrac{dX}{dz}=\dfrac{U_{1}}{z-a_{1}}+\dfrac{U_{2}}{z-a_{2}},\quad U_{1},U_{2}\in M_{2}(\mathbb{C})\). \textbf{Theorem 1}. Let in the differential equation (1) \(\rho_{k},\sigma_{k},k=1,2,\) are the characteristic numbers of matrices \(U_{k};\rho,\sigma\) are the characteristic numbers of the matrix \(U_{1}+U_{2},0<Re(\sigma-\rho)\le1,\alpha_{k}=exp(2\pi i\rho_{k}),\beta_{k}=exp(2\pi i\sigma_{k}),k=1,2,\alpha_{3}=exp(-2\pi i\rho),\beta_{3}=exp2\pi i(1-\sigma),\) and the matrix \(C\) brings the matrix \(U_{1}+U_{2}\) to the Jordanian form. Then, the monodromy matrices of equation (1) can be found by the following formulas: 1) If \(\sigma-\rho\neq0\) and \(\sigma-\rho\neq1\), then \(V_{1}=C\left(\begin{array}{cc} v & c(\alpha_{1}-v)\\ \frac{1}{c}(v-\beta_{1}) & s_{1}-v \end{array}\right)C^{-1},V_{2}=d_{2}C\left(\begin{array}{cc} \beta_{3}(s_{1}-v) & c\alpha_{3}(v-\alpha_{1})\\ \frac{\beta_{3}}{c}(\beta_{1}-v) & \alpha_{3}v \end{array}\right)C^{-1},V_{3}=C\left(\begin{array}{cc} \alpha_{3} & 0\\ 0 & \beta_{3} \end{array}\right)C^{-1}\), where \(v=\frac{d_{2}d_{3}s_{1}-\beta_{3}s_{2}}{\alpha_{3}-\beta_{3}}\). 2) If \(\sigma-\rho=1\), then \(V_{1}=C\left(\begin{array}{cc} cv_{1} & v_{1}\\ -\frac{(cv_{1}-\alpha_{1})(cv_{1}-\beta_{1})}{v_{1}} & s_{1}-cv_{1} \end{array}\right)C^{-1},V_{2}=C\left(\begin{array}{cc} s_{2}-cv_{2} & -v_{2}\\ \frac{(cv_{2}-\alpha_{2})(cv_{2}-\beta_{2})}{v_{2}} & cv_{2} \end{array}\right)C^{-1},V_{3}=C\left(\begin{array}{cc} \alpha_{3} & 0\\ \alpha_{3} & \alpha_{3} \end{array}\right)C^{-1}\), where \(v_{1}=\alpha_{3}d_{1}s_{2}-s_{1},v_{2}=\alpha_{3}d_{2}v_{1}=s_{2}-\alpha_{3}d_{2}s_{1},c\) is an arbitrary constant. As the next simple counter example shows, the second part of the theorem 1 is incorrect. Consider the case when \(U_{1}=\left(\begin{array}{cc} \rho & 0\\ 0 & 0 \end{array}\right)\) and \(U_{2}=\left(\begin{array}{cc} 0 & 0\\ 0 & \rho-1 \end{array}\right)\), then the fundamental matrix of the system will have the form \(X=\left(\begin{array}{cc} (z-a_{1})^{\rho} & 0\\ 0 & (z-a_{2})^{\rho-1} \end{array}\right)\). Therefore, the Galois group, and so the monodromy group, of this equation consists of diagonal matrices. This means that it cannot contain an undiagonalized matrix \(V_{3}\).