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A simple proof of the non-integrability of the first and the second Painlevé equations

Henryk Żołądek (2011)

Banach Center Publications

The first and the second Painlevé equations are explicitly Hamiltonian with time dependent Hamilton function. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems in ℂ⁴. We prove that the latter systems do not have any additional algebraic first integral. In the proof equations in variations with respect to a parameter are used.

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