On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym   equations
  We 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. 
Paolo Lorenzoni and Marco Pedroni