  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.
