  We calculate cross sections of a pair $B_c$ meson production on the basis of
two-photon mechanism from electron-positron annihilation. We investigate the
production cross sections in nonrelativistic approximation and with the account
of relativistic corrections. Relativistic production amplitudes of S-wave pair
pseudoscalar, vector and pseudoscalar+vector $B_c$-mesons are constructed on
the basis of relativistic quark model. Numerical values of the production cross
sections are obtained at different center-of-mass energies. The comparison of
one-photon and two-photon annihilation contributions is presented.
