Non-integrability of the rational Sasano system of type $A^{(2)}_4$

In this paper we study the integrability of the Sasano system of type $A^{(2)}_4$ from the point of view of the Hamiltonian dynamics. We prove rigorously that for these values of the parameters for which the Sasano system of type $A^{(2)}_4$ has a particular rational solution it is not integrable by rational first integrals. By an explicit computation we show that the connected component $G^0$ 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 $SL_2(\mathbb{C})$. This the Morales-Ramis theory leads to a non-integrable result. Moreover using B\"{a}cklund transformations we extend the obtained particular non-integrable result to the entire orbit of the parameters.