Projective structure and integrable geodesic flows on the extension of Bott-Virasoro group

This is a sequel to my paper IHES/M/00/23, triggered from a question posed by Marcel-Ovsienko-Roger in their paper (Lett. Math. Phys. 40 (1997) 31-39). In this paper we show that the multicomponent (or vector) Ito equation, modified dispersive water wave equation and modified dispersionless long wave equation are the geodesic flows with respect to an $L^2$ metric on the semidirect product space ${\widehat {Diff^s(S^1) \bo {C^{\infty}(S^1)}^k}}$, where $Diff^s(S^1)$ is the group of orientation preserving Sobolev $H^s$ diffeomorphisms of the circle. We also study the projective structure associated with the matrix Strum-Liouville operators on the circle.