- \sqrt{\frac{f}{h}}\frac{1}{r^p}
\partial_r\left[ \frac{r^p}{\sqrt{fh}}A_r\right] 
- \frac{1}{r^2}\tilde{\nabla}^i A_i\,.
\label{G}$$ Here $i$ denotes angular variables on the unit $p$-sphere $S^p$ with metric $\tilde{\eta}_{ij}$ and inverse metric $\tilde{\eta}^{ij}$ \[with signature $(+\cdots+)]$, $\tilde{\nabla}_i$ is the associated covariant derivative on $S^p$ and $\tilde{\nabla}^i \equiv \tilde{\eta}^{ij}\tilde{\nabla}_j$. This choice for $K^\mu$ is convenient because the equation for $A_t$ decouples from the other ones. The field equations (\[geneq\]) become $$\begin{aligned}
& & -\frac{1}{f} \partial_t^2 A_t + \sqrt{\frac{f}{h}\,} \frac{1}{r^p}
\partial_r \left[ \frac{r^p }{\sqrt{fh\,}}\partial_r A_t \right] 
+ \frac{1}{r^2}\tilde \nabla^2 A_t = 0 \,,
\label{eqt} \\
& & -\frac{1}{f} \partial_t^2 A_r + \frac{1}{f} \partial_r \left[ 
\sqrt{\frac{f}{h}} \frac{f}{r^p} \partial_r \left( \frac{r^p }{\sqrt{fh\,}} A_r
\right)\right]+ \frac{1}{r^2}\tilde \nabla^2 A_r
+ \frac {1}{f}\partial_r \left( \frac{f}{r^2} \right) \tilde \nabla^i A_i =0\,,
\label{eqr} \\
& & -\frac{1}{f} \partial_t^2 A_i + \frac{r^{2-p}}{\sqrt{fh\,}} 
\partial_r\left( \sqrt{\frac{f}{h}} r^{p-2} \partial_r A_i\right)
- \frac{r^2}{fh} 
\partial_r\left(\frac{f}{r^2}  \right) \partial_i A_r 
+\frac{1}{r^2} \left[ \tilde \nabla ^j (\tilde\nabla_j A_i-\tilde\nabla_i A_j)
 \right.
\nonumber \\
& & \left.
+ \partial_i(\tilde\nabla^j A_j)\right] = 0\,,
\label{eqang}\end{aligned}$$ where $\tilde{\nabla}^2 \equiv \tilde{\eta}_{ij} \tilde\nabla^i\tilde\nabla^j$.

We shall now describe a complete set of solutions $A_\mu^{(\lambda n;\, \omega lm)}$. We assign $\lambda$ the value 0 for what we call the non-physical modes, 1 or 2 for the physical modes and 3 for the pure-gauge modes. (These 