Painleve equations, integrable systems and the stabilizer set of Virasoro orbit

We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on S1, which in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method, we obtain the first integral of the SORE and the results are applied to the study of its Lagrangian and Hamiltonian descriptions. Using these results, we show the existence of a Lagrangian description for SORE, and the Painleve II equation is analyzed.