In this paper, by classical solutions we mean solutions to Fuchsian type meromorphic linear integrable Pfaffian systems dy = Ωy on the complex linear spaces ℂⁿ, n ≥ 1, where y(z) = (y₁(z),..., y_n(z)^T ∈ ℂⁿ is a column vector and Ω is a meromorphic matrix differential 1-form such that Ω = ∑_1≤i<j≤nJ_ij (β)(z_i − z_j)⁻¹ d(z_i − z_j), with constant matrix coefficients Jij(β) depending on complex parameters β = (β₁,..., β₁). Under some constraints on the constant matrix coefficients J_ij (β), the solution components y_i(z), 1 ≤ i ≤ n, can be expressed as integrals of products of powers of linear functions; i.e., they are generalizations of the integral representation of the classical hypergeometric function F(z, a, b, c). Moreover, under some additional constraints on the parameters β, the components of the solutions are hyperelliptic, superelliptic, or polynomial functions. We describe such constraints on the coefficients J_ij(β) of Fuchsian type systems, as well as describe constraints on the sets of matrices (B₁(z),...,B_n(z)) for which the nonlinear Schlesinger equations dB_i(z) = \(-\sum{_{j=1, j\neq i}^n}\)[B_i(z),B_j(z)](z_i − z_j)⁻¹ d(z_i − z_j) reduce to linear integrable Pfaffian systems of the type described above and have solutions of the indicated type.

中文翻译：

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线性Pfaffian系统和三角Schlesinger方程的经典解

在本文中，通过经典的解决方案，我们以富克斯型亚纯线性积Pfaffian系统平均解ð ý =Ω ý在复数线性空间ℂ ^Ñ，Ñ ≥1，其中ÿ（ż）=（Ý ₁（Ž），.. 。，Y _ñ（ż）^Ť ∈ℂ ^ñ是一个列向量和Ω是一个亚纯矩阵微分1形式，使得Ω=Σ _{1 ≤i<j≤n} Ĵ _IJ（β）（ž_我- ž _Ĵ）^{- 1个}d（ž_我- Ž _Ĵ），具有恒定的矩阵系数JIJ（β），这取决于复杂的参数β =（β ₁ ，...，β ₁）。在恒定矩阵系数J _ij（β）的某些约束下，解分量y _i（z）1≤i≤n可以表示为线性函数幂的乘积的积分；即，它们是经典超几何函数F（z，a，b，c）。而且，在对参数β的一些附加约束下，解的分量是超椭圆函数，超椭圆函数或多项式函数。我们描述了对Fuchsian型系统的系数J _ij（β）的这种约束，以及对非线性Schlesinger方程的矩阵集（B ₁（z），...，B _n（z））的约束。d B _i（z）= \（-\ sum {_ {j = 1，j \ neq i} ^ n} \） [ B _i（z），B _j（z）]（z _i - z _j）^-1 d（z _i - z _j）简化为上述类型的线性可积Pfaffian系统，并具有所示类型的解决方案。