z^3 - \sqrt{3} z^8 & 0 & \sqrt{3} z^6 \\ 
0 & 0 & 0 & \sqrt{3} z^5 & -\sqrt{3} z^4 & \sqrt{3} z^7 & -\sqrt{3} z^6 & 0
\end{array}
\right) \, .$$

The null vectors (and resulting Pfaffian system) of this J were found using [*Mathematica*]{} to be

$$\begin{aligned}
\gamma ^{(1)} & = & \Delta ^{-1} [2\sqrt{3}(z^1 z^2 z^4 - (z^1)^2 z^5 + z^1
z^3 z^7 + z^4 z^6 z^7 + z^5 (z^7)^2 + \sqrt{3} z^1 z^7 z^8){\rm d}z^1 
\nonumber \\
& + & 2\sqrt{3}((z^2)^2 z^4 - z^1 z^2 z^5 + z^2 z^3 z^7 + z^5 z^6 z^7 - z^4
(z^7)^2 + \sqrt{3} z^2 z^7 z^8){\rm d}z^2  \nonumber \\
& + & \sqrt{3}(2z^2 z^3 z^4 - 2 z^1 z^3 z^5 + 2 (z^3)^2 z^7 + (z^4)^2 z^7 +
(z^5)^2 z^7 - (z^6)^2 z^7  \nonumber \\
&& - (z^7)^3 + 2 \sqrt{3} z^3 z^7 z^8){\rm d}z^3  \nonumber \\
& + & 2\sqrt{3}(z^2 (z^4)^2 - z^1 z^4 z^5 + 2 z^3 z^4 z^7 + z^1 z^6 z^7 -
z^2 (z^7)^2){\rm d}z^4  \label{SU3-1} \\
& + & 2\sqrt{3}(z^2 z^4 z^5 - z^1 (z^5)^2 + 2 z^3 z^5 z^7 + z^2 z^6 z^7 +
z^1 (z^7)^2){\rm d}z^5  \nonumber \\
& + & 2\sqrt{3}(z^2 