A simple proof of the non-integrability of the first and the second Painlevé equations
Volume 94 / 2011
Banach Center Publications 94 (2011), 295-302
MSC: Primary 34M55; Secondary 37J30.
DOI: 10.4064/bc94-0-20
Abstract
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 $\mathbb{C}^4$. 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.
