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

Neither equation (\[SU3-1\]) nor (\[SU3-2\]) can be integrated by itself. So to obtain two independent Casimir invariants $C_1$ and $C_2,$ we must find exact linear combinations of the equations. For example,

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

This equation is exact, and has the solution

$$\label{SU3-5}
C_1 = (z^1)^2 + (z^2)^2 + (z^3)^2 + (z^4)^2 + (z^5)^2 + (z^6)^2 + (z^7)^2 +
(z^8)^2 \, .$$

Another independent solution can be found by taking the linear combination

$$\begin{aligned}
a \gamma ^{(1)} + b \gamma ^{(2)} & = & (-18 z^4 z^6 - 18 z^5 z^7 - 12 \sqrt{%
3} z^1 z^8){\rm d}z^1  \nonumber \\
&+&(-18 z^5 z^6 + 18 z^4 z^7 - 12 \sqrt{3} z^2 z^8){\rm d}z^2  \nonumber \\
&+&(-9 (z^4)^2 - 9 (z^5)^2 + 9 (z^6)^2 + 9 (z^7)^2 -12 \sqrt{3} z^3 z^8){\rm %
d}z^3  \nonumber \\
&+&(-18 z^3 