Linear integral equations and two-dimensional Toda systems

The direct linearization framework is presented for the two-dimensional (2D) Toda equations associated with the infinite-dimensional Lie algebras (Formula presented.), (Formula presented.), and (Formula presented.), as well as the Kac–Moody algebras (Formula presented.), (Formula presented.), (Formula presented.), and (Formula presented.) for arbitrary integers (Formula presented.), 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.