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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.
Henryk Żołądek. "A simple proof of the non-integrability of the first and the second Painlevé equations." Banach Center Publications 94.1 (2011): 295-302. <http://eudml.org/doc/281688>.
@article{HenrykŻołądek2011, 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 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.}, author = {Henryk Żołądek}, journal = {Banach Center Publications}, keywords = {Painlevé equations; integrability in Liouville-Arnold sense; equation in variations}, language = {eng}, number = {1}, pages = {295-302}, title = {A simple proof of the non-integrability of the first and the second Painlevé equations}, url = {http://eudml.org/doc/281688}, volume = {94}, year = {2011}, }
TY - JOUR AU - Henryk Żołądek TI - A simple proof of the non-integrability of the first and the second Painlevé equations JO - Banach Center Publications PY - 2011 VL - 94 IS - 1 SP - 295 EP - 302 AB - 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. LA - eng KW - Painlevé equations; integrability in Liouville-Arnold sense; equation in variations UR - http://eudml.org/doc/281688 ER -