the following expression at | ⃗ q | ≪ h accurate to within first order in | ⃗ q | The Eq. (25) shows that the frequencies of both branches of the spectrum in the vicinity of the center of the Brillouin zone increase linearly with | ⃗ q | at a nonzero magnetic field. A study of the obtained dispersion relation allows a determination of a stability region for the AFM state, i.e. a region of parameters at which ω $^{2}$( ⃗ q ) becomes negative. The problem of finding the stability region for the AFM state is essentially simplified by the fact that the local minima of the lower branch of the magnon spectrum are located in the center ( ⃗ q$_{(0}$$_{,}$$_{0)}$ = 0) or in the corners ( ⃗ q$_{(1}$$_{,}$$_{1)}$ = π ( ⃗ ε$_{1}$ + ⃗ ε$_{2}$ ) / ( $_{a}$√$_{2)) of the Brillouin zone of the sublattice,}$ see Fig. 5. At these points some coefficients of the Hamiltonian vanish, that makes a simple analytical consideration possible. According to numerical calculations, in the absence of the external magnetic field and at small fields, the minimal value of frequency corresponds to magnons having the maximal value of quasimomentum, ⃗ q$_{(1}$$_{,}$$_{1)}$ = π ( ⃗ ε$_{1}$ + ⃗ ε$_{2}$ ) / ( $_{a}$√$_{2) =}$ π⃗ e$_{x}$/a . Also at this point ω$_{±}$ (1 , 1) = A$_{(1}$$_{,}$$_{1)}$ ± √ D 2 (1 , 1) + h $^{2}$. An analysis of ω $^{2}$( ⃗ q ) in the vicinity of this point gives that at zero field the AFM state is stable only for enough large value of anisotropy, β > β$_{cr}$ , If β > β cr , then with increasing of the field the AFM state loses its stability relative to perturbations with ⃗ q$_{(1}$$_{,}$$_{1)}$ at | h | > h$_{cr}$$_{,}$$_{11}$ = √ A 2 (1 , 1) − D 2 (1 , $_{1)}$, see. Fig. 5. The analysis also shows that for | h | > h$_{cr}$$_{,}$$_{0}$ = √ A 2 0 − F 2 0 the AFM state loses the stabil- ity relative to small ⃗ q perturbations. As it is seen from Fig. 5, both these conditions are important at different values of anisotropy. Thus, the AFM state is stable relative to arbi- trary small perturbations subject to the condition, h < h$_{cr}$ = min { h$_{cr}$$_{,}$$_{0}$, h$_{cr}$$_{,}$$_{11}$ } . For small anisotropy β < β$_{1}$ = 3 . 358 an instability with maximum values of q is developed, whereas at β > β$_{1}$ the AFM state is unstable relative to long-wave perturbations with q ≪ 1 /a . β$_{cr}$ = − 3 2 σ$_{c}$ ( π/a, 0) − 7 − $_{3}$√$_{2}$ 8 σ (0) ≈ 2 . 4532 . (26) 19 C. Stability of AFM state. ω$_{±}$ ( ⃗ q ) = γM √ β $^{$_{+ (3}$√$_{2}$$_{/}$$_{4)}$}$( √$_{2}$ − 1 ) σ (0) · √ β − (1 / 2) ( √$_{2}$ − 1 ) σ (0) + πa | ⃗ q | ± γH. (25)