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

Invariant Solutions of the Yamabe equation on the Koiso-Cao Soliton