PEAKONS, R-MATRIX AND TODA LATTICE

The integrability of a family of Hamiltonian systems, describing in a particular case the motion of N ''peakons'' (special solutions of the so-called Camassa-Holm equation) is established in the framework of th e r-matrix approach, starting from its Lax representation. In the gene ral case, the r-matrix is a dynamical one and has an interesting thoug h complicated structure. However, for a particular choice of the relev ant parameters in the Hamiltonian (the one corresponding to the pure ' 'peakons'' case), the r-matrix becomes essentially constant, and reduc es to the one pertaining to the finite (non-periodic) Toda lattice. In triguing consequences of this property are discussed and an integrable time discretisation is derived.