Stabilizer orbit of Virasoro action and integrable systems

We study the stabilizer orbit of the coadjoint action of the Virasoro algebra on its dual. The vector field associated to the stabilizer orbit is called projective vector field. It was shown by Kirillov that the pair formed by the projective vector field and its square root comprises superalgebra. The Euler-Poincare flow on the "Fermionic" part of the superalgebra yields a pseudospherical type equation. In this paper we express the solutions of the Ermakov-Pinney, Normalized Ermakov-Pinney, Painleve II and C. Neumann system type (0 + 1) dimensional integrable systems in terms of the global and local projective vector field, a vector field leaves fixed a given projective connection.