Homogeneous spaces yielding solutions of the k[S] -hierarchy and its strict version

The k[S] -hierarchy and its strict version are two deformations of the commutative algebra k[S] , k=R or C ; in the N×N -matrices, where S is the matrix of the shift operator. In this paper we show first of all that both deformations correspond to conjugating k[S] with elements from an appropriate group. The dressing matrix of the deformation is unique in the case of the k[S] -hierarchy and it is determined up to a multiple of the identity in the strict case. This uniqueness enables one to prove directly the equivalence of the Lax form of the k[S] -hierarchy with a set of Sato-Wilson equations. The analogue of the Sato-Wilson equations for the strict k[S] -hierarchy always implies the Lax equations of this hierarchy. Both systems are equivalent if the setting one works in, is Cauchy solvable in dimension one. Finally we present a Banach Lie group G( S 2 ) , two subgroups P + (H) and U + (H) of G( S 2 ) , with U + (H)⊂ P + (H) , such that one can construct from the homogeneous spaces G S 2 / P + (H) resp. G( S 2 )/ U + (H) solutions of respectively the k[S] -hierarchy and its strict version.

homogeneous spaces, integrable hierarchies, Lax equations, Sato-Wilson form, wave matrices