I am looking for an example of a smooth Fano $3$-fold $X$ over $\mathbb{C}$, with a non-trival $\mathbb{C}^{\*}$-action, which satisfies the following properties:


1. There is a $\mathbb{C}^{\*}$-action such that the fixed point set is finite.
2. $Aut(X)$ contains no copy of $(\mathbb{C^{\*}})^{2}$.
3. The rank of the Picard group is at least $2$.


(Note that if we relax any of these three conditions, then there are examples. If we relax 1. then we can take $X = \mathbb{P}^{1} \times S$, where $S$ is a del Pezzo surface with discrete automorphism group. If we relax 2. then there are toric examples. If we relax $3.$, then the Mukai-Umemura Fano $3$-fold works. Hence, I believe that there is a reasonable chance that such an example should exist.)



$\mathbb{C}^{*}$-actions on Fano $3$-folds
Choose $X$ to be the blow-up of a smooth quadric $Q\subset \mathbb{P}^4$ at a curve $\Gamma\subset Q$ being a smooth normal rational quartic curve. 


In coordinates, you can for instance choose $\Gamma$ to be the image of 
$$\mathbb{P}^1\to \mathbb{P}^4, [u:v]\mapsto [u^4:u^3v:u^2v^2:uv^3:v^3]$$
and $Q$ to be given by
$$x\_2^2+x\_1x\_3-2x\_0x\_4=0$$
so the action of $\mathbb{C}^\*$ given by $[u:v]\mapsto [u:\xi v]$ extends to a unique action of $\mathbb{C}^\*$ on $\mathbb{P}^4$, which has only $5$ fixed points, four of them being on $Q$ and $2$ of them on $\Gamma$, namely $[1:0:0:0:0]$ and $[0:0:0:0:1]$. You can check in coordinates that the action has finitely many fixed points on the blow-up.


Moreover, the group $\mathrm{Aut}^{\circ}X$ cannot contain $(\mathbb{C}^\*)^2$ as this group would come from an action on $Q$ preserving $\Gamma$. (In fact, we can check that $\mathrm{Aut}^{\circ}X$ is $\mathrm{PGL}\_2$ has index $2$ in $\mathrm{Aut} X$).


