  We review our work on the relation between integrability and
infinite-dimensional algebras. We first consider the question of what sets of
commuting charges can be constructed from the current of a \mbox{\sf U}(1)
Kac-Moody algebra. It emerges that there exists a set $S_n$ of such charges for
each positive integer $n>1$; the corresponding value of the central charge in
the Feigin-Fuchs realization of the stress tensor is $c=13-6n-6/n$. The charges
in each series can be written in terms of the generators of an exceptional
\W-algebra. We show that the \W-algebras that arise in this way are symmetries
of Liouville theory for special values of the coupling. We then exhibit a
relationship between the \nls equation and the KP hierarchy. From this it
follows that there is a relationship between the \nls equation and the algebra
\Wi. These examples provide evidence for our conjecture that the phenomenon of
integrability is intimately linked with properties of infinite dimensional
algebras.
