Geodesic Flow and Two (Super) Component Analog of the Camassa-Holm   Equation
  We derive the 2-component Camassa-Holm equation and corresponding N=1 super generalization as geodesic flows with respect to the $H^1$ metric on the extended Bott-Virasoro and superconformal groups, respectively. 
Partha Guha and Peter J. Olver