  We study the Higgs sector of the next-to-minimal supersymmetric standard
model (NMSSM) with explicit CP violation at the one-loop level, where the
radiative corrections due to the quarks and squarks of the third generation are
taken into account. We expect that, within a reasonable region of the parameter
space of the present model, at least one of five neutral Higgs bosons may be
produced at the future $e^+ e^-$ International Linear Collider (ILC) with
$\sqrt{s} = 500$ GeV, with cross section larger than 12 fb, 15 fb, and 1.5 fb,
respectively, via the Higgs-strahlung process, the $WW$ fusion process, and the
$ZZ$ fusion process. We find that the effect of the CP phase in the present
model yields significant influences upon the production cross sections of the
five neutral Higgs bosons. We also study the decay modes of the five neutral
Higgs bosons to find that their decay widths are similarly affected by the CP
phase. Some of the decay modes in the present model behave differently from
those of the Standard Model.
