z^4 - 18 z^1 z^6 + 18 z^2 z^7 + 6 \sqrt{3} z^4 z^8){\rm d}z^4 
\nonumber \\
&+&(-18 z^3 z^5 - 18 z^2 z^6 - 18 z^1 z^7 + 6 \sqrt{3} z^5 z^8){\rm d}z^5
\label{SU3-6} \\
&+&(-18 z^1 z^4 - 18 z^2 z^5 + 18 z^3 z^6 + 6 \sqrt{3} z^6 z^8){\rm d}z^6 
\nonumber \\
&+&(18 z^2 z^4 - 18 z^1 z^5 + 18 z^3 z^7 + 6 \sqrt{3} z^7 z^8){\rm d}z^7 
\nonumber \\
&+&(-6 \sqrt{3} (z^1)^2 - 6 \sqrt{3} (z^2)^2 - 6 \sqrt{3} (z^3)^2 + 3 \sqrt{3%
} (z^4)^2  \nonumber \\
&& + 3 \sqrt{3} (z^5)^2 + 3 \sqrt{3} (z^6)^2 + 3 \sqrt{3} (z^7)^2 + 6 \sqrt{3%
} (z^8)^2){\rm d}z^8 =0 \, ,  \nonumber\end{aligned}$$

where

$$\begin{aligned}
a & = & -3 \sqrt{3}[-2 (z^1)^2 - 2 (z^2)^2 - 2 (z^3)^2 + (z^4)^2 + (z^5)^2 +
(z^6)^2  \nonumber \\
&& + (z^7)^2 + 2 (z^8)^2],  \label{SU3-7} \\
b & = & 6[3 z^2 z^4 - 3 z^1 z^5 + 3 z^3 z^7 + \sqrt{3} z^7 z^8] \, .
\label{SU3-8}\end{aligned}$$

Integrating equation (\[SU3-6\]) yields the solution

$$\begin{aligned}
C_2 & = & -18 z^1 z^4 z^6 -18 z^1 z^5 z^7 - 6 \sqrt{3} (z^1)^2 z^8 +18 z^2
z^4 z^7 -18 z^2 z^5 z^6  