  In this work, we investigate the degenerate heavy Higgs bosons in the
Next-to-Minimal Supersymmetric Standard Model (NMSSM) by introducing
vector-like particles. Such an extension is well motivated from the top-down
view since some grand unified theories usually predict the existence of singlet
scalars and vector-like particles at weak scale. Under the constraints from the
LHC and dark matter experiments, we find that (1) the null results of searching
for high mass resonances have tightly constrained the parameter space; (2) two
degenerate heavy singlet Higgs bosons $h_2$ and $a_1$ can sizably decay to
$\chi^0_1\chi^0_1$ invisibly. Therefore, search for the monojet events through
the process $gg \to h_2 / a_1 (\to \chi^0_1 \chi^0_1) j$ may further test our
scenario at the future LHC.
