Integrable deformations in the NxN matrices

Inside the algebra $LT_{\mathbb{N}}(R)$ of $\mathbb{N} \times \mathbb{N}$-matrices with coefficients from a commutative algebra $R$ over $k=\mathbb{R}$ or $\mathbb{C}$, that possess only a finite number of nonzero diagonals above the central diagonal, we consider two deformations of commutative Lie subalgebras generated by the $n$-th power $S^{n}, n\geqslant 1,$ 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 $LT_{\mathbb{N}}(R)$. This yields the h$[S^{n}]$-hierarchy and its strict version. Both sets of Lax equations are equivalent to a set of zero curvature equations. To these sets of zero curvature equations we associate two Cauchy problems and present sufficient conditions under which they can be solved. They hold in particular in the formal power series context. Next we introduce two $LT_{\mathbb{N}}(R)$-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.