HIDDEN SYMMETRIES AND LINEARIZATION OF THE MODIFIED PAINLEVE-INCE EQUATION

The linearization and hidden symmetries of the modified Painleve-Ince equation, y'' + sigmayy' + betay3 = 0, where sigma and beta are consta nts, are presented. The linearization of this equation by a nonlocal t ransformation yields a damped (stable) or growing (unstable) harmonic oscillator equation for beta > 0. Hidden symmetries are analyzed by tr ansforming the modified Painleve-Ince equation to a third-order ordina ry differential equation (ODE) which, in general, is invariant under a three-parameter group by a Riccati transformation. A type I hidden sy mmetry is found of a second-order ODE found from the third-order ODE w here a symmetry is lost in the reduction of order by the non-normal su bgroup invariants. A type II hidden symmetry occurs in the third-order ODE because the symmetries of a second-order ODE, reduced from the th ird-order ODE by another set of normal subgroup invariants, are increa sed.