spacetime must be considered physically equivalent. In the case in which $\K^n=S^n$ and the orientation-preserving spatial isometry group is given by $\SO(n+1)$, the Killing vectors are in one-to-one correspondence with the antisymmetric matrices of rank $n+1$, and the transformation of a Killing vector by an isometry is mapped to a conjugate transformation of the corresponding antisymmetric matrix by an element of $\SO(n+1)$. Because the conjugate classes of these anti-symmetric matrices are classified according to their $[(n+1)/2]$ eigenvalues, where $[(n+1)/2]$ is the rank of $\SO(n+1)$, physically distinct exceptional modes are classified with respect to $[(n+1)/2]$ constants. Thus, they have exactly the same number of degrees of freedom as that of the angular momentum parameters of the Myers-Perry solution and the Gibbons-Lü-Page-Pope solution. In fact, we can directly check that these exceptional modes can be obtained by expanding these solutions with respect to the angular momentum parameters. Note that when $\K^n$ is distinct from $S^n$, the parameter $J$ does not represent the angular momentum in the standard sense, although it is still related to a conserved quantity of the system. The number of physical degrees of freedom of the exceptional modes can differ from $[(n+1)/2]$ in that case.

Scalar perturbations {#subsec:BasicEqs:Scalar}
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