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

Geometrical and Dynamical Aspects of Nonlinear Higher-Order Riccati Systems