  We consider the non-trivial Ricci soliton on
$\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ constructed by Koiso and Cao. It is a
K\"ahler metric invariant by the $U(2)$ action on
$\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$. We study its Yamabe equation and
prove it has exactly one $U(2)-$invariant solution up to homothecies.
