A Lagrangian view on complete integrability of the two-component   Camassa-Holm system
  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. 
Jonathan Eckhardt and Katrin Grunert
nlin.SI math.AP