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In this paper we study the integrability of the Hamiltonian system associated with the fourth Painlevé equation. We prove that one two‐parametric family of this Hamiltonian system is not integrable in the sense of the Liouville–Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations, we deduce that the connected component of the unit element of the corresponding differential Galois group is not Abelian. Thus the Morales–Ramis–Simó theory leads to a nonintegrable result. Moreover, combining the new result with our previous one we establish that for all values of the parameters for which the PIV$P_{IV}$ equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in t. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the integrability of the fourth Painlev� equation (PIV) from the point of view of the Hamiltonian dynamics. We prove that the fourth Painlev� equation with parameters a�=�m,�b�=�−2(1�+�2n�+�m) where , is not integrable in the Liouville–Arnold sense 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 the Ziglin–Ramis–Morales-Ruiz–Sim� method yield the non-integrability results. [ABSTRACT FROM AUTHOR]