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

Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions