A dynamical system modelling the mutual interaction of the so-called "peakons" of the Camassa-Holm equation is considered. The system has been recently proved to be endowed with a Lax pair and to be completely integrable by F.Calogero and J.P.Francoise. Here, the complete integrability is demonstrated in the framework of the R-matrix approach. In a special case, the system reveals an intimate connection with the Toda lattice : inspired by that, we have been able to factorize the Lax matrix and to construct an auto-Backlund transformation or, in other words, an integrable time-discretisation of the system.