  This is the Part II 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 the Part I paper [K.~Nakamura,
arXiv:2110.13508 [gr-qc]], we examine the linearized Einstein equations for
even-mode perturbations. We discuss the strategy to solve the linearized
Einstein equations for these even-mode perturbations including $l=0,1$ modes.
Furthermore, we explicitly derive the $l=0,1$ mode solutions to the linearized
Einstein equations in both the vacuum and the non-vacuum cases. We show that
the solutions for $l=0$ mode perturbations includes the additional
Schwarzschild mass parameter perturbation, which is physically reasonable.
Then, we conclude that our proposal of the resolution of the $l=0,1$-mode
problem is physically reasonable due to the realization of the additional
Schwarzschild mass parameter perturbation and the Kerr parameter perturbation
in the Part I paper.
