Multi-poisson approach to the Painlevé equations: From the isospectral deformation to the isomonodromic deformation

A multi-Poisson structure on a Lie algebra g provides a systematic way to construct completely integrable Hamiltonian systems on g expressed in Lax form ∂X_λ/∂t = [X_λ, A_λ] in the sense of the isospectral deformation, where X_λ, A_λ ∈ g depend rationally on the indeterminate λ called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax equation ∂X_λ/∂t = [X_λ, A_λ] + ∂A_λ/∂λ in the sense of the isomonodromic deformation, which exhibits the Painlevé property, is proposed. This method gives a few new Painlevé systems of dimension four.