**Question 1.** Is there a classification of smooth $S^1$-actions with isolated fixed points on the standard $\mathbb CP^3$?

**Question 2.** What if one additionally imposes the condition that the action preserves an almost complex structure homotopic to the standard one? (here we consider almost complex structures up to homotopies through  $S^1$-invariant almost complex structures).
 
Note that non-linear actions on $\mathbb CP^3$ do exist, as is proved by  Ted Petrie, see for example section 4: 

*Smooth $S^1$ actions on homotopy complex projective spaces and related topics* http://www.ams.org/journals/bull/1972-78-02/S0002-9904-1972-12898-2/

I tried to check the articles that cite this article, but was not able to understand if the answer to my questions are known.