\(x^3+y^3+3xy\left(x+y\right)+1-3xy\left(x+y\right)-3xy=0\)
\(\Leftrightarrow\left(x+y\right)^3+1-3xy\left(x+y+1\right)=0\)
\(\Leftrightarrow\left(x+y+1\right)\left(x^2+y^2+2xy-x-y+1\right)-3xy\left(x+y+1\right)=0\)
\(\Leftrightarrow\left(x+y+1\right)\left(x^2+y^2-xy-x-y+1\right)=0\)
\(\Leftrightarrow\left(x+y+1\right)\left[\left(x-\frac{y}{2}-\frac{1}{2}\right)^2+\frac{3}{4}\left(y-1\right)^2\right]=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x+y=-1\\\left\{{}\begin{matrix}x-\frac{y}{2}-\frac{1}{2}=0\\y-1=0\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x+y=-1\\\left\{{}\begin{matrix}x=1\\y=1\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x+y=-1\\x+y=2\end{matrix}\right.\)