  Multipeakons are special solutions to the Camassa-Holm equation described by
an integrable geodesic flow on a Riemannian manifold. We present a
bi-Hamiltonian formulation of the system explicitly and write down formulae for
the associated first integrals. Then we exploit the first integrals and present
a novel approach to the problem of the dissipative prolongations of
multipeakons after the collision time. We prove that an $n$-peakon after a
collision becomes an $n-1$-peakon for which the momentum is preserved.
