  Based on the color-singlet model, we investigate the photoproduction of
$J/\psi$ associated with a $c \bar{c}$ pair with all subprocesses including the
direct, single-resolved, and double-resolved channels. The amplitude squared of
these subprocesses are obtained analytically. By choosing corresponding
parameters, we give theoretical predictions for the $J/\psi$ transverse
momentum distributions both at the LEPII and at the future photon colliders for
these subprocesses. The numerical results show that at the LEPII these
processes can not give enough contributions to account for the experimental
data, and it indicates that the color-octet mechanism may still be needed. At
the photon collider with the laser back scattering photons, the resolved photon
channe will dominate over the direct one in small and moderate $p_t$ regions
with large $\sqrt{s}$. By measuring the $J/\psi$ production associated with a
$c\bar{c}$ pair, this process can be separated from the inclusive $J/\psi$
production and may provide a new chance to test the color-singlet
contributions.
