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