Quadratic Equations and Monodromy Evolving Deformations

In this paper we study a special class of monodromy evolving deformations (MED), which represents Halphen’s quadratic system. In 1996, Chakravarty and Ablowitz [4] showed that a fifth-order equation (DH-V) ω 1 = ω2ω3 − ω1(ω2 + ω3) + φ , ω 2 = ω3ω1 − ω2(ω3 + ω1) + θ , ω 2 = ω3ω1 − ω2(ω3 + ω1)− φθ, (1) φ = ω1(θ − φ) − ω3(θ + φ), θ = −ω2(θ − φ)− ω3(θ + φ), which arises in complex Bianchi IX cosmological models can be represented by MED. The DH-V is solved by the Schwarzian function S(z; 0, 0, a) (three angles of the Schwarzian triangle are 0, 0 and aπ) and a special case of Halphen’s quadratic system. Since generic Schwarzian functions have natural boundary or moving branch points, (1) cannot be obtained as monodromy preserving deformations, because monodromy preserving deformations has the Painlevé property. The system (1) can be represented as the compatibility condition for ∂Y ∂x = μI − (C+x 2 + 2Dx+ C−) P Y, (2)