Invariant solutions of the Yamabe equation on the Koiso-Cao soliton

We consider the non-trivial Ricci soliton g on CP2#CP2‾ constructed by Koiso and Cao. It is a Kähler metric invariant by the U(2) action on CP2#CP2‾. We study the Yamabe equation and prove that it has exactly one U(2)-invariant solution up to homothecies.