We give a symplectic description of the fiber space for each -th Painlevé system ( , , ) which was constructed by K. Okamoto. Hamiltonian function in every chart is a polynomial of the canonical coordinates.
Tohru MATANO. Atusi MATUMIYA. Kyoichi TAKANO. "On some Hamiltonian structures of Painlevé systems, II." J. Math. Soc. Japan 51 (4) 843 - 866, October, 1999. https://doi.org/10.2969/jmsj/05140843