  The direct linearisation framework is presented for the two-dimensional Toda
equations associated with the infinite-dimensional Lie algebras $A_\infty$,
$B_\infty$ and $C_\infty$, as well as the Kac--Moody algebras $A_{r}^{(1)}$,
$A_{2r}^{(2)}$, $C_{r}^{(1)}$ and $D_{r+1}^{(2)}$ for arbitrary integers
$r\in\mathbb{Z}^+$, 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
linearising type solution potentially leading to the universal solution space,
for each class of the two-dimensional Toda system. As particular applications
of this framework to the two-dimensional Toda lattices, we rediscover the Lax
pairs and the adjoint Lax pairs and simultaneously construct the generalised
Cauchy matrix solutions.
