The six Painlevé equations can be rewritten in Hamiltonian forms, with time dependent Hamilton functions. We present a rather new approach to this result, leading to rational Hamilton functions. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems with two degrees of freedom. We realize the Bäcklund transformations of Painlevé equations as symplectic birational transformations in C⁴ and we interpret the cases with classical solutions as the cases of partial integrability of the extended Hamiltonian systems. We prove that the extended Hamiltonian systems do not have any additional algebraic first integral besides known special cases of the third and fifth equations. We also show that the original Painlevé equations admit first integral expressed in elementary functions only in the above special cases. In the proofs we use equations in variations with respect to a parameter and the Liouville's theory of elementary functions.