An integrable hierarchy including the AKNS hierarchy and its strict version

We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. We split the loop space g of gl2 into Lie subalgebras g≥0 and g<0 of all loops with respectively only positive and only strictly negative powers of the loop parameter. We choose a commutative Lie subalgebra C in the whole loop space s of sl2 and represent it as C = C≥0⊕C<0. We deform the Lie subalgebras C≥0 and C<0 by the respective groups corresponding to g<0 and g≥0. Further, we require that the evolution equations of the deformed generators of C≥0 and C<0 have a Lax form determined by the original splitting. We prove that this system of Lax equations is compatible and that the equations are equivalent to a set of zero-curvature relations for the projections of certain products of generators. We also define suitable loop modules and a set of equations in these modules, called the linearization of the system, from which the Lax equations of the hierarchy can be obtained. We give a useful characterization of special elements occurring in the linearization, the so-called wave matrices. We propose a way to construct a rather wide class of solutions of the combined AKNS hierarchy.