  A general 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. Our scheme not only provides a unified perspective to understand
the underlying integrability structure, but also induces a general 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.
