  The QCD amplitudes for particle's production in the quasi-multi-Regge
kinematics with a quark exchange in crossing channels are calculated in the
Born approximation. In particular they are needed to find next-to-leading
corrections to the quark Regge trajectory and to the integral kernel of the
Bethe-Salpeter equation for the t-channel partial wave with fermion quantum
numbers and a negative signature. The gauge-invariant action for the
interaction of the reggeized quarks and gluons with the usual particles is
constructed.
