  The algebraic matrix hierarchy approach based on affine Lie $sl (n)$ algebras
leads to a variety of 1+1 soliton equations. By varying the rank of the
underlying $sl (n)$ algebra as well as its gradation in the affine setting, one
encompasses the set of the soliton equations of the constrained KP hierarchy.
The soliton solutions are then obtained as elements of the orbits of the
dressing transformations constructed in terms of representations of the vertex
operators of the affine $sl (n)$ algebras realized in the unconventional
gradations. Such soliton solutions exhibit non-trivial dependence on the KdV
(odd) time flows and KP (odd and even) time flows which distinguishes them from
the conventional structure of the Darboux-B\"{a}cklund Wronskian solutions of
the constrained KP hierarchy.

Exactly Solvable and Integrable Systems;High Energy Physics - Theory;Exactly Solvable and Integrable Systems