# Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation

@article{Chiba2016MultiPoissonAT, title={Multi-Poisson Approach to the Painlev{\'e} Equations: from the Isospectral Deformation to the Isomonodromic Deformation}, author={Hayato Chiba}, journal={Symmetry Integrability and Geometry-methods and Applications}, year={2016}, volume={13}, pages={025} }

A multi-Poisson structure on a Lie algebra $\mathfrak{g}$ provides a systematic way to construct completely integrable Hamiltonian systems on $\mathfrak{g}$ expressed in Lax form $\partial X_\lambda /\partial t = [X_\lambda , A_\lambda ]$ in the sense of the isospectral deformation, where $X_\lambda , A_\lambda \in \mathfrak{g}$ depend rationally on the indeterminate $\lambda $ called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax…

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