1 0 Ju n 20 02 The Camassa - Holm hierarchy , N - dimensional integrable systems , r - matrix structure , and algebro - geometric solution

This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We find that the well-known CH equation is included in the negative order CH hierarchy while a Dym type equation is included in the posi-