Inside the algebra LT N (R) of N×N-matrices with coefficients from a commutative algebra R over k = R or C, that possess only a finite number of nonzero diagonals above the central diagonal, we consider two deformations of commutative Lie subalgebras generated by the nth power S n , n ⩾ 1, of the matrix S of the shift operator and a maximal commutative subalgebra h of gl n (k), where the evolution equations of the deformed generators are determined by a set of Lax equations, each corresponding to a different decomposition of LT N (R). This yields the h[S n ]-hierarchy and its strict version. We show that both sets of Lax equations are equivalent to a set of zero curvature equations. Next we introduce two Cauchy problems linked with these sets of zero curvature equations and present sufficient conditions under which they can be solved. Moreover, we show that these conditions hold in the formal power series context. Next we introduce two LT N (R)-models, one for each hierarchy, a set of equations in each module and special vectors satisfying these equations from which the Lax equations of each hierarchy can be derived. We conclude by presenting a functional analytic context in which these special vectors can be constructed. Thus one obtains solutions of both hierarchies.

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.