  We show how the change from Eulerian to Lagrangian coordinates for the
two-component Camassa-Holm system can be understood in terms of certain
reparametrizations of the underlying isospectral problem. The respective
coordinates correspond to different normalizations of an associated first order
system. In particular, we will see that the two-component Camassa-Holm system
in Lagrangian variables is completely integrable as well.
