  We investigate the direct photoproduction of $J/\psi+c\bar{c}$ at $ep$
colliders with different center of mass energies in this paper. For the
color-singlet part we take into account the contribution from QED subprocess as
well as the QCD one. Our results show that the QED subprocess not only enhances
the differential cross section in the large $p_t$ region but also gives a more
transverse $J/\psi$ with $p_t$ increase, which is different from the almost
unpolarized $J/\psi$ predicted by the QCD subprocess. The contribution from
color octet subprocesses enhances the differential cross section by a factor of
5 in large $p_t$ region and predicts an even more transverse $J/\psi$ in the
medium and large $p_t$ region. The sensitivity of this process to the
color-octet matrix element $\langle O^{J/\psi}(^3S_1^8)\rangle$ may give great
help in studying the matrix element.
