For the purposes of this question, a spherical lens is the interesection of two spherical half-spaces in $\mathbb{S}^3$. (A spherical half-space is the intersection of $\mathbb{S}^3$ with a half-space in $\mathbb R^4$ whose boundary contains the origin.) My question is:
Is there an action of $S^1$ on $\mathbb{S}^3$ by isometries that preserves a given spherical lens?
I feel like the answer should be yes, but my intuition is very Euclidean and the coordinates are too dense for my liking. Topologically, the two spherical half-spaces are 3-balls, so their boundaries are 2-spheres, interesecting in a circle. The spheres have the same radius, so the lens is symmetric. If I try to picture this in $\mathbb R^3$, I can see that there is a line passing through the centres of the top and bottom faces, and any rotation about this line preserves the lens. But this is a Euclidean rotation in $\mathbb R^3$—does a similar 'rotation' exist for $\mathbb{S}^3$?