  On the basis of the perturbative QCD and the relativistic quark model we
investigate the relativistic and bound state effects in the production
processes of a pair of $S$-wave doubly heavy mesons with opposite charge
conjugation consisting of $b$ and $c$ quarks. All possible relativistic
corrections in the production amplitude including the terms connected with the
transformation law of the bound state wave function to the reference frame of
the moving pseudoscalar ${\cal P}-$ and vector ${\cal V}-$ mesons are taken
into account. We obtain a growth of the cross section for the reaction
$e^++e^-\to J/\Psi+ \eta_c$ due to considered effects by a factor $2\div 2.5$
in the range of the center-of-mass energy $\sqrt{s}=6\div 12$ GeV.
