The direct linearization framework is presented for the two-dimensional (2D) Toda equations associated with the infinite-dimensional Lie algebras , , and , as well as the Kac–Moody algebras , , , and for arbitrary integers , from the aspect of a set of linear integral equations in a certain form. Such a scheme not only provides a unified perspective to understand the underlying integrability structure, but also induces the direct linearizing type solution potentially leading to the universal solution space, for each class of the 2D Toda system. As particular applications of this framework to the 2D Toda lattices, we rediscover the Lax pairs and the adjoint Lax pairs and simultaneously construct the generalized Cauchy matrix solutions.

中文翻译：

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线性积分方程和二维 Toda 系统

直接线性化框架被呈现用于与无限维李代数相关的二维（2D）户方程，和，以及所述KAC-Moody代数，，，和为任意整数，从一定形式的一组线性积分方程来看。这种方案不仅提供了一个统一的视角来理解底层的可积性结构，而且还为二维 Toda 系统的每一类引入了可能导致通用解空间的直接线性化类型解。作为该框架在 2D Toda 格子上的特殊应用，我们重新发现了 Lax 对和伴随 Lax 对，并同时构建了广义柯西矩阵解。