  Minimal extensions of the Standard Model (SM), such as the so-called Two
Higgs Doublet Model (2HDM), can possess several accidental discrete symmetries
whose spontaneous breakdown in the early universe usually lead to the formation
of domain walls. We extend an earlier work [arXiv:2006.13273] on this topic by
studying in more detail the analytic properties of electrically charged and
CP-violating kink solutions in the $Z_{2}$-symmetric 2HDM. We derive the
complete set of equations of motion that describe the 1D spatial profile of
both the 2HDM vacuum parameters and the would-be Goldstone bosons $G^{1,2,3}$
of the SM. These equations are then solved numerically using the gradient flow
technique, and the results of our analysis are presented in different
parametrizations of the Higgs doublets. In particular, we show analytically how
an electrically charged profile should arise in 1D kink solutions when
asymmetric boundary conditions are imposed on the Goldstone mode $G^{2}$ at
spatial infinities, i.e. as ${x\to \pm \infty}$. If asymmetric boundary
conditions are selected at ${x\to \pm \infty}$ for the Goldstone mode $G^{3}$
or the longitudinal mode $\theta$ corresponding to a would-be massive photon,
the derived kink solutions are then shown to exhibit CP violation. Possible
cosmological implications of the electrically charged and CP-violating domain
walls in the 2HDM are discussed.
