Geometrical and Dynamical Aspects of Nonlinear Higher-Order Riccati Systems

We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on $S^1$, whichn in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method we obtain the first-integral of SORE and the results are applied to the study of its Lagrangian and Hamiltonian description. We unveil the relation between the Darboux polynomials and master symmetries associated to second-order Riccati. Using these results we show the existence of a Lagrangian description for the related system, and the Painlev\'e II equation is analysed.