Seminarium KMMF "Teoria Dwoistości"

The six Painlevé equations can be rewritten in Hamiltonian forms, with timedependent Hamilton functions. We present a rather new approach to this result,leading to rational Hamilton functions. By a natural extension of the phase spaceone gets corresponding autonomous Hamiltonian systems with two degrees of freedom.We realize the Bäcklund transformations ofPainlevé equations as symplectic birational transformations in IC^4 and we interpretthe cases with classical solutions as the cases of partial integrability of theextended Hamiltonian systems. We prove that the extended Hamiltonian systems do nothave any additional algebraic firstintegral besides known special cases of the third and fifth equations. We also showthat the original Painlevé equations admit first integral expressed in elementaryfunctions only in the above special cases. In the proofs we use equations invariations with respect to a parameter and the Liouville's theory of elementaryfunctions.