extremum, which is a saddle point in the center of the Brillouin zone is found. II. MODEL DESCRIPTION. For uniform magnetization inside of each particle the state of the dot array is described by the full magnetic moment of each dot, ⃗ m$_{⃗}$ $_{l}$, ∣ ⃗ m$_{⃗}$ l ∣ = m$_{0}$ placed in the square lattice sites ⃗ l = a ( ⃗ e$_{x}$l$_{x}$ + ⃗ e$_{y}$l$_{y}$ ), where l$_{x}$, l$_{y}$ are integers. The system Hamiltonian describing the dipole interaction of dots subject to an external magnetic field ⃗ H$_{0}$ and taking into account the uniaxial magnetic anisotropy for each dot reads 35 ˆ H = 1 2 ∑ ⃗ l = ⃗ l ′ ⃗ m$_{⃗}$ $_{l}$⃗ l ′ − 3( ⃗ m$_{⃗}$ $_{l}$⃗ ν )( ⃗ m$_{⃗}$ l ′ ⃗ ν $^{′}$) ∣ ∣$^{⃗}$ l − ⃗ l ′ ∣ ∣$^{3}$ − − ∑ ⃗ l [ 1 2 κ · ( ⃗ m$_{⃗}$ $_{l}$⃗ e$_{z}$ ) 2 + ⃗ m$_{⃗}$ l ⃗ H$_{0}$ ] . (1) Here ⃗ ν = ( ⃗ l − ⃗ l $^{′}$) / ∣ ∣$^{⃗}$ l − ⃗ l ′ ∣ , κ is the anisotropy constant for a given dot, which is assumed to be uniaxial with a easy axis ⃗ e$_{z}$ , perpendicular to the dot array plane. For dots made of soft magnetic material, the anisotropy is associated with the dot shape ( κ > 0 or κ < 0 for dots oblate or oblong along ⃗ e$_{z}$ , respectively). For dots made of highly anisotropic magnetic materials, 7,8 the crystalline anisotropy of the material can provide some contribution to κ . The state of a single dot can be characterized by perpendicular magnetization ( ⃗ m$_{z}$ = ± m$_{0}$⃗ e$_{z}$ ). However, in the following only the most symmetric case will be considered where the external field ⃗ H$_{0}$ is perpendicular to the array plane. Although oscillations of magnetization in the dot can be treated in a purely classical manner, it is convenient to employ the operator approach. 33 For dots with homogeneous magnetization it is natural to introduce magnon creation and annihilation operators for the total magnetic moment. To find a dispersion law in the linear approximation, it is sufficient to use the formulae, 35 below we will put Lande factor g = 2, which in this approximation are similar to the familiar Holstein-Primakoff or Dyson-Maleev representations. 35 Here µ$_{B}$ is the Bohr magneton, 1, 2, m$_{3}$$_{,}$$_{⃗}$ l = m$_{0}$ − gµ$_{B}$a † ⃗ $_{l}$a$_{⃗}$ $_{l}$, m$_{1}$$_{,}$$_{⃗}$ l = ( a † ⃗ l + a$_{⃗}$ $_{l}$) √ gm$_{0}$µ$_{B}$/ 2 , (2) m$_{2}$$_{,}$$_{⃗}$ l = i ( a † ⃗ l − a$_{⃗}$ $_{l}$) gm$_{0}$µ$_{B}$/ 2 , 5