e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}}  ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot 
\nonumber\\
\nonumber\\
\cdot \tilde{f}_{1}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;-
\nonumber\\
\nonumber\\
-\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{3}\gamma_{2}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) }  \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}}  ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot 
\nonumber\\
\nonumber\\
\cdot \tilde{f}_{2}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;+
\nonumber\\
\nonumber\\
+\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ (\gamma_{1}^2 + \gamma_{2}^2)} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) }  \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}}  ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot 
\nonumber\\
\nonumber\\
\cdot \tilde{f}_{3}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;+
\nonumber\\
\nonumber\\
+\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t}\biggl[  \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}}  ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})}
 \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad
\nonumber\\
\nonumber\\
\cdot \; u_{3}^{0}(\tilde x_{1},\tilde x_{2}, \tilde x_{3})\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3} \;= 
\nonumber\\
\nonumber\\
=\; S_{31}(\tilde{f}_{1})\;+\; S_{32}(\tilde{f}_{2})\;+\; S_{33}(\tilde{f}_{3})\;+\;B(u_{3}^0) ,\;\;\; (\ref{A8})
\quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \end{aligned}$$

Here $S_{11}(),\; S_{12}(),\; S_{13}(),\; S_{21}(),\; S_{22}(),\; S_{23}(),\; S_{31}(),\; S_{32}(),\; S_{33}(),\; B()\; $ are the integral operators.

$$S_{12}()\;= \;S_{21}()$$$$S_{13}()\;= \;S_{31}()$$$$S_{23}()\;= \;S_{32}()$$

It follows from $(\ref{eqn160})\;-\; (\ref{eqn162})\;$ that 