  The transform under study is defined by integration of functions over spheres
centered on a sphere. Such transform is of interest due to its applications in
analysis and (thermoacoustic) tomography. The range of this transform has been
described recently. Besides natural smoothness and support conditions, two type
of conditions were involved: orthogonality condition and, in even dimension,
moment condition. However, the moment condition was later derived from the
other ones, which implies that orthogonality condition completely characterizes
the range. We present a direct proof of this fact by proving existence of
global extension for solutions of a certain boundary value problem for Darboux
equation, associated with the spherical means. This extendibility phenomenon
seems to be of independent interest.
