  We express AKNS hierarchies, admitting reductions to matrix NLS and matrix
mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a
universal result in this framework quickly generates an infinite family of
exact solutions, including e.g. the matrix solitons in the focusing NLS case.
Exploiting a general Miura transformation, we recover the generalized
Heisenberg magnet hierarchy and establish a corresponding solution formula for
it. Simply by exchanging the roles of the two derivations of the bidifferential
graded algebra, we recover "negative flows", leading to an extension of the
respective hierarchy. In this way we also meet a matrix and vector version of
the short pulse equation and also the sine-Gordon equation. For these equations
corresponding solution formulas are also derived. In all these cases the
solutions are parametrized in terms of matrix data that have to satisfy a
certain Sylvester equation.
