Non-integrability of the Painlevé IV equation in the Liouville-Arnold sense and Stokes phenomena

In this paper we study the integrability of the Hamiltonian system associated to 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 Galois grou is not Abelian. Thus the Morales-Ramis-Simó theory leads to a non-integrable result. Moreover, combining the new result with our previous one we establish that for allvalues of the parameters for which the $P_{IV}$ equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in $t$.