  Let $\Sigma$ be an axially symmetric, smooth, closed hypersurface in $\Bbb
R^{n + 1}$ with a simply connected interior which is contained inside the unit
sphere $\Bbb S^{n}$. For a continuous function $f$, which is defined on $\Bbb
S^{n}$, the main goal of this paper is to characterize the support of $f$ in
case where its integrals vanish on subspheres obtained by intersecting $\Bbb
S^{n}$ with the tangent hyperplanes of a certain subdomain
$\mathcal{U}\subset\Sigma$ of $\Sigma$. We show that the support of $f$ can be
characterized in case where its integrals also vanish on subspheres obtained by
intersecting $\Bbb S^{n}$ with hyperplanes obtained by infinitesimal
perturbations of the tangent hyperplanes of $\mathcal{U}$ and where
$\mathcal{U}$ satisfies some regularity condition which implies local
convexity.
