dependence is placed in the points of the type of X (1,0), where ⃗ k = ± π⃗ e$_{x}$/a or ⃗ k = ± π⃗ e$_{y}$/a . Thus for small λ four saddle points are symmetrically located inside the Brillouin zone on the directions such as (0,1) with | ⃗ k | < π/a . A local minimum leads to existence of an additional jump of the density of states having the form D ( ω ) = C$_{1}$ + C$_{2}$ · Θ ( ω − ω$_{lm}$ ) inside the frequency band for λ < λ$_{c}$ , see Fig. 3b, whereas the saddle points produce standard logarithmic singularities. It is also seen that at the critical value of the parameter λ = λ$_{c}$ ≃ 3 . 6, where two Van Hove singularities emerge at some critical value, ω$_{c}$ , the density of states shows a new type of singularity which is stronger than the standard logarithmic singularity (compare Fig. 3c and Figs. 3a,b). To calculate this dependence, one have to take into account the terms of order ( ⃗ k − ⃗ k$_{0}$ ) $^{4}$. It is remarked here that the value of the parameter λ = ω$_{0}$/ω$_{int}$ depends on the external magnetic field and can be changed continuously for the same sample during the experiment, thus, the observation of this singularity is possible. On the other hand, the non-standard (linear in k ) dispersion near the gap frequency ω$_{0}$ produce much weaker singularities in the density of states. In particular, for ω ≥ ω$_{0}$ simple calculations give Δ D ( ω ) = C · ( ω − ω$_{0}$ )Θ ( ω − ω$_{0}$ ) instead of finite jump. This singularity is clearly seen for all values of λ (see Fig.3). In the previous section we investigated the states of the array in which all dots are in the same magnetic state (FM ground state of array). Next consider the AFM structure where the transition from FM to AFM states can be controlled by a weak external magnetic field. The spectral analysis of the dot array in the AFM state is of much interest because this transition from FM to AFM states provides the possibility to tune the properties of collective modes in the system. The method developed here might be easily generalized for the case of simple AFM states, which might be described within the framework of a few sublattices. To do that, it is necessary to split the array onto different sublattices with the same dot state in each sublattice, and introduce different Bose operators for each sublattice. IV. CHESSBOARD ANTIFERROMAGNETIC STATE. 13 D ( ω ) = C | ω − ω$_{c}$ | 1 / 4 , (14)