  Recovering a function from its integrals over circular cones recently gained
significance because of its relevance to novel medical imaging technologies
such emission tomography using Compton cameras. In this paper we investigate
the case where the vertices of the cones of integration are restricted to a
sphere in $n$-dimensional space and symmetry axes are orthogonal to the sphere.
We show invertibility of the considered transform and develop an inversion
method based on series expansion and reduction to a system of one-dimensional
integral equations of generalized Abel type. Because the arising kernels do not
satisfy standard assumptions, we also develop a uniqueness result for
generalized Abel equations where the kernel has zeros on the diagonal. Finally,
we demonstrate how to numerically implement our inversion method and present
numerical results.
