in Refs. , but some are new. In order for solutions to these basic equations to represent regular perturbations of black hole spacetimes, an appropriate boundary condition at the horizon and an asymptotic condition at infinity should be satisfied. These boundary conditions are specified in §3, and it is shown that the exceptional vector perturbations indeed do satisfy them. Then, in the subsequent three sections, it is proved that, except for the exceptional vector perturbations and the trivial scalar perturbations corresponding to variations of the background parameters, there exists no stationary solution satisfying the boundary conditions to the basic equations for the asymptotically flat, de Sitter and anti-de Sitter cases, respectively. Section 7 is devoted to concluding remarks.

Basic Equations for Stationary Perturbations
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Our analysis of black hole perturbations fully utilises the gauge-invariant formulation developed in Refs. and . In this section, we briefly summarise the basic concepts of that formulation and give master equations for stationary perturbations.

Background spacetime
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In the present paper, we consider the background spacetime whose metric has the form where $g_{ab}(y)dy^a dy^b$ is the static metric of a two-dimensional spacetime $\N^2$, and $d\sigma_n^2=\gamma_{ij}(z)dz^i dz^j$ is the metric of an $n$-dimensional complete, compact Einstein 