  We developed general approach to the calculation of power-law infrared
asymptotics of spin-spin correlation functions in the Kitaev honeycomb model
with different types of perturbations. We have shown that in order to find
these correlation functions, one can perform averaging of some bilinear forms
composed out of free Majorana fermions, and we presented the method for
explicit calculation of these fermionic densities. We demonstrated how to
derive an effective Hamiltonian for the Majorana fermions, including the
effects of perturbations. For specific application of the general theory, we
have studied the effect of the Dzyaloshinskii-Moriya anisotropic spin-spin
interaction; we demonstrated that it leads, already in the second order over
its relative magnitude $D/K$, to a power-law spin correlation functions, and
calculated dynamical spin structure factor of the system. We have shown that an
external magnetic field $h$ in presence of the DM interaction, opens a gap in
the excitation spectrum of magnitude $\Delta \propto D h$.
