by the integral, ∑ ⃗ l → ∫ rdrdχ , where r and χ are the polar coordinates. Integration over χ can be done exactly through the Bessel function J$_{0}$ ( qr ) and the non-analytical part of the sum σ ( ⃗ k ) at small | ⃗ k | is determined by the integral I = 2 π ∞ ∫ 0 ( dr/r $^{2}$) J$_{0}$ ( qr ) (1 − exp( − αr $^{2}$)), which is convergent both as r → 0 and r → ∞ . It is convenient to integrate by parts, which in the case we are interested in ( α → 0 ) yields I = 2 π ∞ ∫ 0 ( dr/r ) J$_{1}$ ( qr ) = 2 π | q | . From here we arrive at the expression, used in (7) For the complex sum σ$_{c}$ ( ⃗ k ) near the origin ⃗ k = ⃗ q , | ⃗ q | ≪ 1 /a the same approach gives the double integral, where α is the angle between ⃗ q and the x axis. Here the integral over χ can be done, which gives σ$_{c}$ = 2 πe − 2 iα ∫ ( dr/r $^{2}$) J$_{2}$ ( qr ). Further, for the integral over r , which is convergent both as r → 0 due to asymptotic J$_{2}$ ( z ) → (1 / 2)( z/ 2) 2 as z → 0 and for r → ∞ , it is not necessary to include the regularization multiplier exp( − αr $^{2}$) like in (27). Finally we arrive at the following expression Thus the relation between σ ( ⃗ k ) and σ$_{c}$ ( ⃗ k ), σ ( ⃗ k ) → 3 | σ$_{c}$ ( ⃗ k ) | as ⃗ k → 0, which is of great importance for the spectral analysis is asymptotically exact at small values of | ⃗ k | . This result is in accordance with the numerical calculation. Let us discuss now the calculation of the series σ ( ⃗ k ) and σ$_{c}$ ( ⃗ k ) near the second symmetric point ⃗ k$_{0}$ of the type of (1,1). Here for both series the values of D$_{ij}$ = [ ∂ $^{2}$σ ( ⃗ k )/ ∂k$_{i}$∂k$_{j}$ ]$_{⃗}$ k = ⃗ k 0 are described by convergent alternating series like ∑ P$_{2}$$_{p}$ ( ⃗ l ) · ( − 1) l$_{x}$ + l$_{y}$ ( l 2 x + l 2 y ) − (2 p +1) / $^{2}$, where P$_{2}$$_{p}$ ( ⃗ l ) is the polynomial in l$_{x}$, l$_{y}$ of degree 2 p , p = 1 and p = 2 for components D$_{ij}$ in the case of σ ( ⃗ k ) and σ$_{c}$ ( ⃗ k ), respectively. By using symmetry relations, one can demonstrate that the value of the function σ ( ⃗ k ) at ⃗ k of type (1,1) is described only by one sum, D = − ∑ ( − 1) l$_{x}$ + l$_{y}$ ( l 2 x + l 2 y ) − 1 / $^{2}$, the value of which can be easily found numerically, D ∼ = 1 . 61554 . As a result the following expression is obtained σ ( ⃗ k ) ∼ = σ ( ⃗ k$_{0}$ ) + 1 4 Da $^{2}$| ⃗ k − ⃗ k$_{0}$ | $^{2}$. σ$_{c}$ ( ⃗ k ) = 2 π 3 a ( k$_{x}$ − ik$_{y}$ ) 2 | ⃗ k | as ⃗ k → 0 . σ$_{c}$ ( ⃗ q ) = ∞ ∫ 0 dr r 2 · 2 π ∫ 0 dχe − 2 iχ · e iqr cos( χ − α $^{)}$, σ ( ⃗ k ) = σ (0) − 2 πa | ⃗ k | at ⃗ k → 0 . 28