  We study the class of shallow water equations of Camassa and Holm derived
from the Lagrangian: $ L= \int \left( \frac{1}{2} (\varphi_{xxx}-\varphi_{x}
)\varphi_{t} - {1 \over 2} {(\varphi_{x})^{3}} - {1 \over
2}\varphi_{x}(\varphi_{xx})^{2} - {1 \over 2} \kappa \varphi_{x}^{2} \right)
dx, $
