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 $\mathfrak{g}$ of $gl_2$ into a Lie subalgebras $\mathfrak{g}_{\ge0}$ and $\mathfrak{g}_{<0}$ 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 $\mathfrak{s}$ of $sl_2$ and represent it as $C=C_{\ge0}\oplus C_{<0}$. We deform the Lie subalgebras $C_{\ge0}$ and $C_{<0}$ by the respective groups corresponding to $\mathfrak{g}_{<0}$ and $\mathfrak{g}_{\ge0}$. Further, we require that the evolution equations of the deformed generators of $C_{\ge0}$ 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.