TITLE: Extremal Fano with non constant scalar curvature vs Kaehler-Einstein Fano manifolds 
QUESTION [6 upvotes]: The first Hirzebruch surface $F_1$ does not admit any Kaehler metric of constant scalar curvature. Yet it admits an extremal metric representing each Kaehler class as shown by Calabi. The two points blow-up of $\mathbb{CP}^2$ behaves similarly.
Do you know any other example of Fano manifold such that:

 The anti-canonical class does not contain any Kaehler-Einstein metric.
 The anti-canonical class contains an extremal metric.

REPLY [2 votes]: In fact, I am asking the question because I just proved that there are certain deformations of the Mukai-Umemura $3$-fold that belong to Tian's family and admit extremal metrics in the anti-canonical Kaehler classes. However they do not admit any Kaehler-Einstein metric according to the celebrated result of Tian. The remarkable thing is that these examples are arbitrarily small complex deformations of the Mukai-Umemura $3$-fold which does admit a Kaehler-Einstein metric by Donaldson's computation of the $\alpha$-invariant.
I would like to know how much this example of phenomenon is original, and if there are others of this type.