On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa–Holm Equation

Different gauge copies of the Ablowitz–Kaup–Newell–Segur (AKNS) model labeled by an angle $\theta$ are constructed and then reduced to the two-component Camassa–Holm model. Only three different independent classes of reductions are encountered corresponding to the angle $\theta$ being 0, $\pi/2$ or taking any value in the interval $0<\theta<\pi/2$. This construction induces Bäcklund transformations between solutions of the two-component Camassa–Holm model associated with different classes of reduction.