2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/213079Different gauge copies of the Ablowitz-Kaup-Newell-Segur (AKNS) model labeled by an angle $θ$ 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 $θ$ being 0, $π/2$ or taking any value in the interval $0<θ<π/2$. This construction induces Bäcklund transformations between solutions of the two-component Camassa--Holm model associated with different classes of reduction.This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/High Energy Physics - TheoryMathematical PhysicsExactly Solvable and Integrable Systemstext