We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size $(p\times p)$ are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference $q$, the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference $q$, for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the $(2\times 2)$-case, we obtain explicit sequences of rational solutions and of one-parameter families of rational solutions of Painlevé VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems.

我们研究了在任意大小的未知矩阵的情况下的施莱辛格偏微分方程组 $(磷\times 磷)$是三角形的，每个矩阵的特征值形成一个等差级数，有理差$q$，对所有矩阵都一样。我们表明，这样的系统拥有一系列通过超椭圆曲线黎曼曲面上的亚纯微分周期表示的解决方案。我们确定差值$q$，为此，我们的解决方案导致了Schlesinger系统的显式多项式或有理解。作为应用$(2\times 2)$情况下，我们获得PainlevéVI方程的有理解和一参数族有理解的显式序列。使用类似的方法，我们提供特定卡尼尔系统的代数解。