in the general network Fig. \[Fig:kink\_network\] in terms of the vector potential requires application of the gauge field parametrization suggested in a series of papers by Y.M. Cho [@Cho1; @Cho2], S. Shabanov [@shabanov1; @shabanov2], L.D. Faddeev and A. J. Niemi [@Faddeev] and, recently, by K.-I. Kondo [@Kondo]. In this parameterization the Abelian part ${\hat V}_\mu (x)$ of the gauge field ${\hat A}_\mu (x)$ is separated manifestly, $$\begin{aligned}
\label{eqnsFaddeevNiemKondo} 
 {\hat A}_\mu (x) &=& {\hat V}_\mu (x) + {\hat X}_\mu (x), \, 
{\hat V}_\mu (x) = {\hat B}_\mu (x) + {\hat C}_\mu (x), \\
 {\hat B}_\mu (x) &=& [n^aA^a_\mu (x)]\hat{n} (x)=B_\mu(x)\hat{n}(x), \nonumber \\
 {\hat C}_\mu (x) &=& g^{-1}\partial_\mu \hat{n}(x)\times \hat{n}(x), \nonumber\\
 {\hat X}_\mu (x) &=& g^{-1}{\hat n}(x) \times \left( \partial_\mu {\hat n}(x) + g {\hat A}_\mu (x) \times {\hat n}(x) \right), \nonumber\end{aligned}$$ where ${\hat A}_\mu (x) = A^a_\mu (x) t^a$, ${\hat n} (x) = n_a (x) t^a$, $n^a n^a = 1$, and $$\begin{aligned}
{\partial_\mu\hat n}\times {\hat n} = i f^{abc}\partial_\mu n^a n^b t^c,\,
      \, [t^a,t^b]=if^{abc}t^c.\end{aligned}$$ The field ${\hat V}_\mu $ is seen as the Abelian field in the sense that $[{\hat V}_\mu (x),{\hat V}_\nu (x)]=0$. The color vector field $n^a(x)$ may be used for detailed description of the thin domain wall junctions 