# Is there extremal metric on toric Fano manifolds which futaki invariant is nonzero?

According the work by Wang & Zhu, on toric Fano manifolds there exists Kahler-Ricci solitons, if futaki=0 there exist CSCK metric, but if futaki invariant is not vanished, what about extremal metrics? Is there some counterexample?

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Here is an example where the Futaki invariant does not vanish, but with extremal metrics. The Futaki invariant on $Bl_p\mathbb{P}^2$, the blow up of $\mathbb{P}^2$, which is toric, does not vanish (Canonical Metrics in Kaehler geometry by Tian, Example 3,10). On the other hand, $Bl_p\mathbb{P}^2$ admits an extremal metric in every Kaehler class (I believe this is in Calabi's Extremal Kaehler Metrics). Is this what you are looking for?