  Finite rational $\cw$ algebras are very natural structures appearing in coset
constructions when a Kac-Moody subalgebra is factored out. In this letter we
address the problem of relating these algebras to integrable hierarchies of
equations, by showing how to associate to a rational $\cw$ algebra its
corresponding hierarchy. We work out two examples: the $sl(2)/U(1)$ coset,
leading to the Non-Linear Schr\"{o}dinger hierarchy, and the $U(1)$ coset of
the Polyakov-Bershadsky $\cw$ algebra, leading to a $3$-field representation of
the KP hierarchy already encountered in the literature. In such examples a
rational algebra appears as algebra of constraints when reducing a KP hierarchy
to a finite field representation. This fact arises the natural question whether
rational algebras are always associated to such reductions and whether a
classification of rational algebras can lead to a classification of the
integrable hierarchies.
