I am looking for a reference to the following claims:

1) Any compact group (connected or not)  acting  on S^2 is differentiably conjugate to a linear action. This must be classical.

2) A circle S^1 acting on  RP^3 (and supposedly any spherical space form) is
differentiably conjugate to a linear action.
This is probably true for every compact group acting on a 3-dimensional spherical space form?

Wolfgang Ziller