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

Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part III: -- Realization of exact solutions