Non-Integrability of the Sasano System of Type $D_5^{(1)}$ and Stokes Phenomena

In 2006, Y. Sasano proposed higher-order Painlevé systems, which admit affine Weyl group symmetry of type $D_l^{(1)}$, $l=4,5,6,\dots$. In this paper, we study the integrability of a four-dimensional Painlevé system, which has symmetry under the extended affine Weyl group $\tilde{W}\left(D_5^{(1)}\right)$ and which we call the Sasano system of type $D_5^{(1)}$. We prove that one family of the Sasano system of type $D_5^{(1)}$ is not integrable by rational first integrals. We describe Stokes phenomena relative to a subsystem of the second normal variational equations. This approach allows us to compute in an explicit way the corresponding differential Galois group and therefore to determine whether the connected component of its unit element is not Abelian. Applying the Morales-Ramis-Simó theory, we establish a non-integrable result.