Proving that the only glide reflection which maps more than one line to itself on a sphere is an antipodal map

Question

I know that in Spherical geometry, a rotation is the same as a translation. So a glide reflection is the same as a rotation-reflection or translation-reflection. Also, geodesics in $S^2$, are great circles and if the points are antipodal, then there are infinite number of great circles between them. But I'm not sure how to go about the proof for this question.

Answer 1

If a symmetry of the sphere fixes 2 lines, then it must either fix or interchange their 2 points of intersection. That should narrow down the possibilities.