2 where momentum γ ( − k ) = − γ ( k ) and specific for each non- centrosymmetric crystal structure $^{9,17}$, all the values such as single electron energy velocity v$_{αβ}$ ( k ) = ∂ξ$_{αβ}$ ( k ) /∂ k , the inverse effective mass ( m − 1 ij )$_{αβ}$ = ∂ $^{2}$ξ$_{αβ}$ ( k ) /∂k$_{i}$∂k$_{j}$ , the Green func- tions G$_{αβ}$ ( τ$_{1}$, k ; τ$_{2}$, k $^{′}$) = −〈 T$_{τ}$ a$_{k}$$_{α}$ ( τ$_{1}$ ) a † k ′ $_{β}$( τ$_{2}$ ) 〉 and F$_{αβ}$ ( τ$_{1}$, k ; τ$_{2}$, k $^{′}$) = 〈 T$_{τ}$ a$_{k}$$_{α}$ ( τ$_{1}$ ) a$_{−}$$_{k}$ ′ $_{β}$( τ$_{2}$ ) 〉 are matrices in the spin space. Taking this in mind, we obtain The functions ˜ Δ$_{λ}$( k ) are the gaps in the λ -band quasipar- ticle spectrum in superconducting state. In the simplest model with BCS pairing interaction v$_{g}$ ( k , k $^{′}$) = − V$_{g}$ , the gap functions are the same in both bands: ˜ Δ$_{+}$( k ) = ˜ Δ$_{−}$( k ) = Δ and we deal with pure singlet pairing $^{18}$. Transforming the Green functions using eqn. (9) into the band representation $^{19}$, we obtain The transposed matrix of velocity is determined as ˆ v $^{t}$( − k ) = ∂ ˆ ξ $^{t}$( − k ) /∂ k . The arguments of the zero field Green functions are denoted as K$_{±}$ = (Ω$_{m}$ ± ω$_{n}$/ 2 , k ± q / 2). The Matsubara frequencies take the values Ω$_{m}$ = π (2 m + 1 − n ) T and ω$_{n}$ = 2 πnT . and G$_{λ}$ ( ω$_{n}$, k ) = − i ¯ hω$_{n}$ + ξ$_{λ}$ ¯ h $^{2}$ω 2 n + ξ 2 λ + | ˜ Δ$_{λ}$( k ) | $_{2}$, F$_{λ}$ ( ω$_{n}$, k ) = t$_{λ}$ ( k ) ˜ Δ$_{λ}$( k ) ¯ h $^{2}$ω 2 n + ξ 2 λ + | ˜ Δ$_{λ}$( k ) | $_{2}$, (11) t$_{λ}$ ( k ) = − λ γ$_{x}$ ( k ) − iγ$_{y}$ ( k ) √ γ 2 $_{x}$( k ) + γ 2 $_{y}$( k ) . ξ$_{αβ}$ ( k ) = ( ε ( k ) − µ ) δ$_{αβ}$ + γ ( k ) σ $_{αβ}$, (5) j$_{i}$ ( ω$_{n}$, q ) = − e 2 c T r [ ˆ m − 1 ij ˆ n$_{e}$ + ∫ d $^{3}$k (2 π ¯ h ) $_{3}$T ∞ ∑ m = −∞ { ˆ v$_{i}$ ( k ) ˆ G $^{(0)}$( K$_{+}$ )ˆ v$_{j}$ ( k ) ˆ G $^{(0)}$( K$_{−}$ ) +ˆ v$_{i}$ ( k ) ˆ F $^{(0)}$( K$_{+}$ )ˆ v t $_{j}$( − k ) ˆ F $^{+(0)}$( K$_{−}$ ) } ] A$_{j}$ ( ω$_{n}$, q ) . (6) T r { ˆ v$_{i}$ ( k ) ˆ G $^{(0)}$( K$_{+}$ )ˆ v$_{j}$ ( k ) ˆ G $^{(0)}$( K$_{−}$ ) +ˆ v$_{i}$ ( k ) ˆ F $^{(0)}$( K$_{+}$ )ˆ v t $_{j}$( − k ) ˆ F $^{+(0)}$( K$_{−}$ ) } = v$_{++}$$_{,i}$ ( k ) G$_{+}$v$_{++}$$_{,j}$ ( k ) G$_{+}$ + v$_{++}$$_{,i}$ ( k ) F$_{+}$v$_{++}$$_{,j}$ ( − k ) F † + + v$_{−−}$$_{,i}$ ( k ) G$_{−}$v$_{−−}$$_{,j}$ ( k ) G$_{−}$ + v$_{−−}$$_{,i}$ ( k ) F$_{−}$v$_{−}$$_{+}$$_{,j}$ ( − k ) F † + + v$_{+}$$_{−}$$_{,i}$ ( k ) G$_{−}$v$_{−}$$_{+}$$_{,j}$ ( k ) G$_{+}$ + v$_{+}$$_{−}$$_{,i}$ ( k ) F$_{−}$v$_{+}$$_{−}$$_{,j}$ ( − k ) F † + + v$_{−}$$_{+}$$_{,i}$ ( k ) G$_{+}$v$_{+}$$_{−}$$_{,j}$ ( k ) G$_{−}$ + v$_{−}$$_{+}$$_{,i}$ ( k ) F$_{+}$v$_{−}$$_{+}$$_{,j}$ ( − k ) F † $_{−}$(12) H$_{0}$ = ∑ k ξ$_{αβ}$ ( k ) a † k $_{α}$a$_{k}$$_{β}$ = ∑ k ,λ = ± ξ$_{λ}$ ( k ) c † k $_{λ}$c$_{k}$$_{λ}$ (7) ξ$_{λ}$ ( k ) = ε ( k ) − µ + λ | γ ( k ) | , (8) v$_{λλ}$ ′ ( ± k ) = u † $_{λα}$( ± k ) v$_{αβ}$ ( ± k ) u$_{βλ}$ ′ ( ± k ) = ∂ξ$_{0}$ ( ± k ) ∂ k δ$_{λλ}$ ′ + ∂γ$_{l}$ ( ± k ) ∂ k τ$_{l,λλ}$ ′ ( ± k ) , (13) as the trace of the matrices in eqn. (6). For the brevity, we omit here the arguments of the Green functions. They are the same as in the upper two lines. The matrix ve- locity in the band representation is One can pass from the spin to the band representation, where the one-particle Hamiltonian has diagonal form. Here, the band energies are such that two Fermi surfaces are determined by equations ξ$_{λ}$ ( k ) = 0. The difference of the band energies 2 | γ ( k$_{F}$ ) | characterizes the intensity of the spin-orbital coupling. The Fermi momentum with γ = 0 is determined by the equation ε ( k$_{F}$ ) = ε$_{F}$ . The diagonalization is made by the following transfor- mation where ˆ τ ( k ) = ˆ u $^{†}$( k ) ˆ σ ˆ u ( k ) are hermitian matrices. We have neglected 20 the difference between ˆ u ( k ) and ˆ u ( k ± q / 2). The explicit expressions for the ˆ τ ( k ) matrices are with the coefficients where ˆ γ = γ / | γ | , γ$_{⊥}$ = √ γ 2 x + γ 2 $_{y}$. All the diagonal ele- ments of these matrices are odd functions of the momen- tum components. Correspondigly, the products of the diagonal components of velocity matrices (13) are even functions of momentum. Hence, these terms in eqn. (12) The zero field Green functions in the band representa- tion are diagonal and have the following form: 9 √ 2 | γ | forming a unitary matrix ˆ u ( k ). u$_{↑}$$_{λ}$ ( k ) = √ | γ | + λγ$_{z}$ 2 | γ | , u$_{↓}$$_{λ}$ ( k ) = λ γ$_{x}$ + iγ$_{y}$ 2 | γ | ( | γ | + λγ$_{z}$ ) , G (0) λλ ′ ( ω$_{n}$, k ) = δ$_{λλ}$ ′ G$_{λ}$ ( ω$_{n}$, k ) , F (0) λλ ′ ( ω$_{n}$, k ) = δ$_{λλ}$ ′ F$_{λ}$ ( ω$_{n}$, k ) , (10) ˆ τ$_{x}$ = ( ˆ γ$_{x}$ − γ$_{x}$ ˆ γ$_{z}$ + iγ$_{y}$ γ ⊥ − γ$_{x}$ ˆ γ$_{z}$ − iγ$_{y}$ γ ⊥ − ˆ γ$_{x}$ ) , ˆ τ$_{y}$ = ( ˆ γ$_{y}$ − γ$_{y}$ ˆ γ$_{z}$ − iγ$_{x}$ γ ⊥ − γ$_{y}$ ˆ γ$_{z}$ + iγ$_{x}$ γ ⊥ − ˆ γ$_{y}$ ) , (14) ˆ τ$_{z}$ = ( ˆ γ$_{z}$ γ ⊥ γ γ ⊥ γ − ˆ γ$_{z}$ ) , a$_{k}$$_{α}$ = ∑ λ = ± u$_{αλ}$ ( k ) c$_{k}$$_{λ}$, (9)