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On the Lax pairs of the sixth Painleve' equation

Robert Conte
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Abstract

The dependence of the sixth equation of Painleve' on its four parameters $(2 α,-2 β,2 γ,1-2 δ) =(θ_{\infty}^2,θ_{0}^2,θ_{1}^2,θ_{x}^2)$ is holomorphic, therefore one expects all its Lax pairs to display such a dependence. This is indeed the case of the second order scalar ``Lax'' pair of Fuchs, but the second order matrix Lax pair of Jimbo and Miwa presents a meromorphic dependence on $θ_\infty$ (and a holomorphic dependence on the three other $θ_j$). We analyze the reason for this feature and make suggestions to suppress it.

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Robert Conte. 2007-01-24. On the Lax pairs of the sixth Painleve' equation. https://arxiv.org/abs/nlin/0701049

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