  We consider the semidirect product of diffeomorphisms of the circle
$D={Diff}_+(S^1)$ and $C^{\infty}(S^{1}, {\bf R})$ functions, classify its
coadjoint orbits and prove the integrability of hamiltonian (Generalized
Dispersive Water Waves (DWW) and KdV-type) systems related to corresponding Lie
algebra centrally extended by Kac-Moody, Virasoro and semidirect product
cocycles with arbitrary coefficients.
