  This is the Part III paper of our series of papers on a gauge-invariant
perturbation theory on the Schwarzschild background spacetime. After reviewing
our general framework of the gauge-invariant perturbation theory and the
proposal on the gauge-invariant treatments for $l=0,1$ mode perturbations on
the Schwarzschild background spacetime in [K.~Nakamura, arXiv:2110.13508
[gr-qc]], we examine the problem whether the $l=0,1$ even-mode solutions
derived in the Part II paper [K.~Nakamura, arXiv:2110.13512 [gr-qc]] are
physically reasonable, or not. We consider the linearized versions of the
Lema\^itre-Tolman-Bondi solution and the non-rotating C-metric. As the result,
we show that our derived even-mode solutions to the linearized Einstein
equations actually realize above two linearized solutions. This fact supports
that our derived solutions are physically reasonable, which implies that our
proposal on the gauge-invariant treatments for $l=0,1$ mode perturbations are
also physically reasonable. We also briefly summarize our conclusions of our
series of papers.
