Local two-radii theorems on the multi-dimensional sphere

Consider those functions on the $n$-dimensional sphere that have zero integrals over all geodesic balls with centres in a given set $E$. We obtain a description of such functions in the case when $E$ is a geodesic sphere on $\mathbb S^n$. We also find a criterion for the existence of non-zero functions with this property in the case when the set of centres is the union of two geodesic spheres. We obtain analogues of these results for quasi-analytic classes of functions.