and $\K^n$ is expressed as $\K^n=T^p\times 
\K'$, where $\K'$ is a Ricci flat space with no isometry, there exist $p$ independent covariantly constant vector fields providing harmonic vectors with $k_T^2=0$. These modes and those with $k_T^2=n-1$ for $K=1$ comprise the exceptional modes discussed below.

First, for generic modes satisfying $k_V^2>(n-1)K$, the linearised Einstein equations can be written in terms of the gauge-invariant variables as where From the $a=t$ component of , we find that $F^r=0$ for stationary perturbations, and becomes trivial. Hence, the Einstein equations are equivalent to Note that stationary vector perturbations are not static.

Next, for exceptional modes with $k_V^2=(n-1)K$, $\VHB_{ij}$ vanishes. (The factor $1/k_V$ in the definition of $\VHB_{ij}$ is introduced just for convenience and is not essential.) For these modes, $H_T$ does not appear, and perturbations are described by $f_a$ alone. Because $f_a$ transforms under the gauge transformation $\bar\delta 
z^i=L\VHB^i$ as $\bar\delta f_a=-rD_a L$, $f_r$ can be set to zero through such a gauge transformation. In this gauge, the Einstein equations are given by with $m_V=0$ and $F^t=f^t$. This equation can be easily solved, and the general solution is given by From this, we have where $C$ is an integration constant.

The gauge condition $f_r=0$ does 