gauge-invariant quantities constructed from these expansion coefficients: where As shown in Ref. , the linearised Einstein equations for stationary scalar perturbations can be written in terms of these gauge-invariant variables as where As discussed in Ref. , we can reduce this set of equations to a second-order ODE in various ways. For example, the master variable $\Phi$ introduced in Ref. for generic perturbations can be written as a complicated linear combination of $X$ and $Y$ and satisfies a second-order ODE for stationary perturbations. However, this equation is not useful in the analysis of stationary perturbations, because its effective potential is not positive definite in the case of a non-vanishing cosmological constant for generic values of $n$. Further, it is not easy to determine the asymptotic behaviour of $F$ and $F^a_b$, because their expressions in terms of $\Phi$ are rather complicated. Therefore, in the present paper, we utilise second-order ODEs for $X$ and $Y$, which turn out to have structures convenient for the investigation of the uniqueness issue.

The second-order ODE for $Y$ can be easily obtained by eliminating $X$ from with the help of . The result can be expressed as where Similarly, by eliminating $Y$, we obtain the following 