2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/80788We show that the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym hierarchies can be obtained by applying a reduction process to a simple Poisson pair defined on the loop algebra of $\mathfrak{sl}(2,\mathbb{R})$. The reduction process is a bi-Hamiltonian reduction, that can be canonically performed on every bi-Hamiltonian manifold.11 pages, LaTeX; minor changes, to appear in Int. Math. Res. NotExactly Solvable and Integrable SystemsDifferential Geometrytext