New Integrable Hierarchies from Vertex Operator Representations of Polynomial Lie Algebras

We give a representation–theoretic interpretation of recent discovered coupled soliton equations using vertex operators construction of affinization of not simple but quadratic Lie algebras. In this setup we are able to obtain new integrable hierarchies coupled to each Drinfeld–Sokolov of $A$, $B$, $C$, $D$ hierarchies and to construct their soliton solutions.