Nonintegrability of coupled Painlevé systems with affine Weyl group symmetry of type \(A_4^{(2)}\)

We study the integrability of coupled Painlevé systems with affine Weyl group symmetry of type \(A^{(2)}_4\) $A_4^{(2)}$ , which in short we call the Sasano system of type \(A^{(2)}_4\) $A_4^{(2)}$ , from the point of view of the Hamiltonian dynamics. We prove rigorously that for all values of the parameters for which the Sasano system of type \(A^{(2)}_4\) $A_4^{(2)}$ has a particular rational solution it is not integrable by rational first integrals. By an explicit computation we show that the connected component \(G^0\) $G^0$ of the unit element of the differential Galois group of the normal variational equations along a simple particular rational solution is a direct product of two groups \(\textrm{SL}_2({{\mathbb {C}}})\) ${\text{SL}}_2(\mathbb{C})$ . Applying the Morales–Ramis theory we provide a nonintegrable system. Moreover, using Bäcklund transformations we extend the obtained particular result to an orbit of the parameters.