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Riccati Chain, Higher Order Painlevé Type Equations and Stabilizer Set of Virasoro Orbit
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 the projective vector field and the equation associated to this is called the projective vector field equation. At first we study the Riccati and higher Riccati equations associated to this equation. We obtain the solutions of these special higher Riccati equations in terms of the solutions of ordinary Riccati equation. We also derive Painlevé II equation ($\alpha = 2$) from the second order Riccati equation.
Using the geometrical relation between the projective vector field equation and Hill's equation we obtain the solutions of various anharmonic oscillators. Solutions of the Ermakov-Pinney equation, Kummer-Schwarz equation Emden-Fowler and Painlevé II are given in terms of global projective connection. In the second half of the paper we derive generalized Chazy equation for dihedral triangle case, Chazy class XII equation and Painlevé II ($\alpha = - 1/2$) from the second and third order Riccati equations. The relation between the Riccati and the projective vector field equations is explored via invariant methods.