The spectrum behavior essentially depends on the external magnetic field value. The important and quite interesting (see below) case of small fields is determined by the behavior at h = 0. Therefore, we begin with the study of asymptotics of the spectrum in the absence of an external magnetic field. Because of the long-range nature of the dipole interaction, the magnon spectrum has specific peculiarities in the center of the Brillouin zone, therefore it is reasonable to start with an analysis of the long wave limit k → 0. To first order in | ⃗ k | the sums take the form (7) and after simple algebra one obtains the asymptotic form of the dispersion relation at h = 0, This expression describes a linear growth of frequency of the upper branch with increasing q ≡ | ⃗ q | in the vicinity of the center of Brillouin zone. On the other hand, for the frequency of the lower branch ω$_{−}$ ( ⃗ q ) the terms linear in ⃗ q are canceled out, and one can expect that for this branch the dependence on the components of the vector ⃗ q should be parabolic. The numerical data presented in Fig. 5 indicate this. However, there is also a sharp anisotropy of the dependence ω$_{−}$ ( ⃗ q ); namely, in the vicinity of the point ⃗ q = 0 the frequency ω$_{−}$ ( ⃗ q ) increases with a q = | ⃗ q | , if ⃗ q is parallel to one of directions (1,0), and ω$_{−}$ ( ⃗ q ) decreases if ⃗ q is parallel to (1,1). Note by virtue of forth order symmetry of the point ⃗ q = 0 such behavior is forbidden for the quadratic terms. The anisotropy in the expansion of any analytical function ω ( ⃗ q ) in the ⃗ q components can appear only due to invariants of the fourth order, like ( q 4 x + q 4 $_{y}$) or q 2 $_{x}$q 2 $_{y}$. Actually, again a non-analyticity, caused by a slow convergence of the dipolar sums, takes place here. Our analysis shows, in the vicinity of the point ⃗ q = 0 the dispersion relation can be approximated by the expression with the values of the coefficients, α ≃ 0.654 and β ≃ 1 . 8. Thus, in the zero field case a new type of singular behavior of ω ( ⃗ q ), namely, the presence of specific saddle point forth-fold symmetry, appears in the lower magnon branch. In the case of nonzero external magnetic field in the vicinity of the point q → 0 one gets ω ( ⃗ q ) ≈ ω$_{0}$ + α ( q 4 x + q 4 $_{y}$) − 2 βq 2 $_{x}$q 2 y | ⃗ q | 2 , (24) ω$_{±}$ ( ⃗ q ) = γM √ β + ( $^{$_{3}$√$_{2}$$_{/}$$_{4}$}$) ( √$_{2}$ − 1 ) σ (0) $_{·}$√ β − (1 / 2) ( √$_{2}$ − 1 ) σ (0) + πa | ⃗ q | (1 ± 1) . (23) 18