  In 2006 Y. Sasano constructed higher order Painlev\'{e} systems which admit
affine Weyl group symmetry of type $D^{(1)}_l, l=4, 5, 6, \ldots$. Every such a
system is a Hamiltonian one with a coupled Hamiltonian associated to the
Painlev\'{e} V equation when $l$ is an odd or to the Painlev\'{e} VI equation
when $l$ is an even. In this paper we study the integrability of a
four-dimensional Painlev\'{e} system which has symmetry under the extended
affine Weyl group $\widetilde{W}(D^{(1)}_5)$ which we call the Sasano system of
type $D^{(1)}_5$. We prove that one non-singular for the B\"{a}cklund
transformation family of the Sasano system of type $D^{(1)}_5$ is not
integrable by meromorphic first integrals which are rational functions in $t$.
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 threfore to deduce that the
connected component of its unit element is not Abelian. In conclusion the
Morales-Ramis-Sim\'{o} theory leads to the non-integrable result.
