Non-integrability of the Sasano system of type $A^{(2)}_5$

The Sasano sytem of type $A^{(2)}_5$ is a four-dimensional non-linear system of ordinary differential equations, which has an affine Weyl group of symmetries of type $A^{(2)}_5$. It is also a tipe dependent Hamiltonian system, which can be considered as coupled Painlev\'e III systems. In this paper, utilizing the Morales-Ramis-Sim\'o theory for integrability of Hamiltonian systems, we prove rigorously that for all values of the parameters, for which the Sasano system of type $A^{(2)}_5$ admits a particular rational solution, it is non-integrable by rational first integrals.