I am a PhD student in Physics. This problem is motivated by representing a spin-$j$ state by Majorana's stellar representation. There are $2j$ points (Majorana stars) on the two-dimensional sphere (Bloch sphere) where the wave function vanishes. Hence, a spin state corresponds to a configuration of points on the sphere. If the spin is placed in an external magnetic field, the configuration of points undergoes rigid rotation.
For a general cyclic evolution, the stars will move around the sphere (like wrapping a plastic bag around a ball), and finally come back to their original position. Therefore, the mathematical question is: how to classify continuous maps from $S^2$ to $S^2$ with $n$ fixed points? Since the homotopy group $π_2(S^2)$ is $Z$, I guess the answer will be a subgroup of $Z$.
For a general reference, see e.g. https://physics.aps.org/articles/v5/65
Any explanation or reference will be appreciated.