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In this paper we consider the sixth Painlevé equation as a Hamiltonian system depending on parameters. We prove the non-integrability of two families of the Painlevé VI equations, i.e. non-existence of meromorphic first integrals. For this we utilize the differential Galois theory and the Ziglin-Ramis-Morales-Ruiz method.

In this paper we are concerned with the integrability of the fifth Painlevé equation ( ) from the point of view of the Hamiltonian dynamics. We prove that the Painlevé equation (2) with parameters for arbitrary complex (and more generally with parameters related by Bäclund transformations) is non integrable by means of meromorphic first integrals. We explicitly compute formal and analytic invariants of the second variational equations which generate topologically the differential Galois group. In this way our calculations and Ziglin-Ramis-Morales-Ruiz-Simó method yield to the non-integrable results.

We study an irregular singularity of Poincaré rank 1 at the origin of a certain third-order linear solvable homogeneous ODE. We perturb the equation by introducing a small parameter ε ∈ (R + , 0) (ε < 1), which causes the splitting of the irregular singularity into two finite Fuchsian singularities. We show that when the solutions of the perturbed equation contain logarithmic terms, the Stokes matrices of the initial equation are limits of the part of the monodromy matrices around the finite resonant Fuchsian singularities of the perturbed equation.

We prove non-existence of an additional rational integral for the second Painlevé equation (P II ) considered as a Hamiltonian system using Morales -Ramis theory.