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