not remove the gauge degree of freedom completely and leaves a residual gauge freedom such that $L=L(t)$. If we require that $f_t$ be independent of time, $L(t)$ is restricted to the form $L(t)=C' t$, where $C'$ is a constant. Under this gauge transformation, the above integration constant $C$ changes to $C-C'$. Hence, we can set this constant to zero through a gauge transformation. However, as discussed in the next section, this residual gauge degree of freedom plays an important role in proving the regularity of the above solution. In any case, this argument shows that the physical degrees of freedom of these exceptional modes can be parametrised by the single constant $J$ for each mode. As shown in Appendix \[Appendix:ExceptionalModes\], we can regard each of these modes as representing a rotational perturbation of a static black hole, and the parameter $J$ represents the total angular momentum of the perturbation.

Because the vector harmonics satisfying $k_V^2=(n-1)K$ are in one-to-one correspondence with the Killing vector fields of $\K^n$, these solutions form a linear space isomorphic to the linear space of Killing vector fields of $\K^n$. However, they are not all physically distinct, because two solutions related by an isometry of the background 