We strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to $X$, and $\widehat{K}$ contains no analytic discs.

We extend a solution method used for the one-dimensional Toda lattice in [1], [2] to the two-dimensional Toda lattice. The idea is

1. to study the lattice not with values in $\mathbb{C}$ but in the Banach algebra ${\cal L}$ of bounded operators and

2. to derive solutions of the original lattice ($\mathbb{C}$-solutions) by applying a functional $\tau$ to the ${\cal L}$-solutions constructed in 1.

In the second part we focus on applications. We start by rederiving line-solitons and briefly discuss discrete resonance phenomena. Moreover, we are able to find conditions under which it is possible to superpose even countably many line-solitons.