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.
