A system with an inequality

From $x+y=5$: $y=5-x$. Then
$$x^2+(5-x)^2\le13\iff2x^2-10x+25\le13\iff2x^2-10x+12\le0\iff x^2-5x+6\le0.$$
This is $(x-2)(x-3)\le0$, so $x\in[2,3]$. Integer $x\in\{2,3\}$.

Solutions. $(2,3)$ and $(3,2)$.

Conclusion. $(x,y)\in\{(2,3),(3,2)\}$.