  The ground and first excited states of the hybrid charmonium ${\bar c} c g$,
with non-exotic quantum numbers $J^{PC}=0^{-+}$, $1^{--}$ and $1^{++}$ are
investigated using quenched lattice QCD. They are completely ignored in the
literature, only because their ground states are degenerate with $\eta_c$,
$J/\psi$, and $\chi_{c1}$, and are difficult to be distinguished from these
conventional charmonium mesons in experiment. However, we observe strong
gluonic radial excitations in the first excited states; We predict that their
masses are 4.352(225)GeV, 4.379(149)GeV and 7.315(257)GeV, completely different
from the first excited states of the corresponding conventional charmonium.
Their relevance to the recent discovery of the Y(4260) state and future
experimental search for other states are also discussed.
