Аннотация:
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.
