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:

There is a $\mathbb{C}^{*}$-action such that the fixed point set is finite.

$Aut(X)$ contains no copy of $(\mathbb{C^{*}})^{2}$.

- 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.)