  We present and theoretically report the influence of a class of
near-parity-time-(PT-) symmetric potentials with spectral filtering parameter
$\alpha_2$ and nonlinear gain-loss coefficient $\beta_2$ on solitons in the
complex Ginzburg-Landau (CGL) equation. The potentials do not admit
entirely-real linear spectra any more due to the existence of coefficients
$\alpha_2$ or $\beta_2$. However, we find that most stable exact solitons can
exist in the second quadrant of the $(\alpha_2, \beta_2)$ space, including on
the corresponding axes. More intriguingly, the centrosymmetric two points in
the $(\alpha_2, \beta_2)$ space possess imaginary-axis (longitudinal-axis)
symmetric linear-stability spectra. Furthermore, an unstable nonlinear mode can
be excited to another stable nonlinear mode by the adiabatic change of
$\alpha_2$ and $\beta_2$. Other fascinating properties associated with the
exact solitons are also examined in detail, such as the interactions and energy
flux. These results are useful for the related experimental designs and
applications.
