  The spherical domains $S^d_\beta$ with conical singularities are a convenient
arena for studying the properties of tensor Laplacians on arbitrary manifolds
with such a kind of singular points. In this paper the vector Laplacian on
$S^d_\beta$ is considered and its spectrum is calculated exactly for any
dimension $d$. This enables one to find the Schwinger-DeWitt coefficients of
this operator by using the residues of the $\zeta$-function. In particular, the
second coefficient, defining the conformal anomaly, is explicitly calculated on
$S^d_\beta$ and its generalization to arbitrary manifolds is found. As an
application of this result, the standard renormalization of the one-loop
effective action of gauge fields is demonstrated to be sufficient to remove the
ultraviolet divergences up to the first order in the conical deficit angle.
