对于二维自治微分系统，首次积分的存在完全决定其相图。本文考虑三次微分系统 \begin{align} \left\{ \begin{array}{l} \frac{\mathrm{d} x}{\mathrm{d} t} = \,y+ \mathrm{a_{20}}\, x^2 + \mathrm{a_{11}}\, x\, y+\mathrm{a_{02}}\, y^2 + \mathrm{a_{30}}\, x^3 \\ \quad \quad + \mathrm{a_{21}}\, x^2\, y + \mathrm{a_{12}}\, x\, y^2 + \mathrm{a_{03}}\, y^3 \\ \frac{\mathrm{d} y}{\mathrm{d} t} =-x+ \mathrm{b_{20}}\, x^2 + b_{11}\, x\, y+\mathrm{b_{02} }\, y^2+\mathrm{b_{30}}\, x^3 \\ \quad \quad + \mathrm{b_{21}}\, x^2\, y + \mathrm{b_{12}}\, x\, y^2 + \mathrm{b_{03}}\, y^3 \end{array} \right. \end{align} 其中$\mathrm{b_{02}}=\mathrm{b_{20}}+\mathrm{a_{11}},\;\mathrm{b_{11}}=\mathrm{a_{20}}-\mathrm{a_{02}},\;\mathrm{b_{03}}=\mathrm{b_{21}}-\mathrm{a_{30}}+ \mathrm{a_{12}},\;\mathrm{b_{12}}=\mathrm{b_{30}}-\mathrm{a_{03}}+\mathrm{a_{21}}$。 通过因式分解系统的$n$次超切曲线求解不变代数曲线和指数因子，构造系统的达布首次积分．