  Let $\mathfrak{t}$ be a commutative Lie subalgebra of ${\rm
sl}_{n}(\mathbb{C})$ of maximal dimension. We consider in this paper three
spaces of $\mathfrak{t}$-loops that each get deformed in a different way. We
require that the deformed generators of each of them evolve w.r.t. the
commuting flows they generate according to a certain, different set of Lax
equations. This leads to three integrable hierarchies: the $({\rm
sl}_{n}(\mathbb{C}), \mathfrak{t})$-hierarchy, its strict version and the
combined $({\rm sl}_{n}(\mathbb{C}), \mathfrak{t})$-hierarchy. For $n=2$ and
$\mathfrak{t}$ the diagonal matrices, the $({\rm sl}_{2}(\mathbb{C}),
\mathfrak{t})$-hierarchy is the AKNS-hierarchy. We treat their interrelations
and show that all three have a zero curvature form. Furthermore, we discuss
their linearization and we conclude by giving the construction of a large class
of solutions.
