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 $k[S]$, $k=\mathbb{R}$ or $\mathbb{C},$ in the $\mathbb{N} \times \mathbb{N}$-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(\mathcal{S}_{2}),$ two subgroups $ P_{+}(H)$ and $ U_{+}(H)$ of $G(\mathcal{S}_{2}),$ with $ U_{+}(H) \subset P_{+}(H),$ such that one can construct from the homogeneous spaces $G(\mathcal{S}_{2})/ P_{+}(H)$ resp. $G(\mathcal{S}_{2})/U_{+}(H)$ solutions of respectively the $k[S]$-hierarchy and its strict version.