Related to sum of three squares and this question.
Let $l_1,l_2,l_3 \in \mathbb{Z}[x,y]$ and $n \in \mathbb{Z}$.
Assume that $n$ is not a cube and not twice cube.
Let $f=l_1(x,y)^3+l_2(x,y)^3+l_3(x,y)^3-n$.
Is it possible $f$ to factors as $f(x,y)=q(x,y)g(x,y)$ where $\deg q=2$ and $q=0$ is absolutely irreducible and have infinitely many integral points, giving infinitely many representations of $n$ as sum of three cubes?
For $n= -1$ this is possible:
$l_1=2*x - 1;l_2=-x + y;l_3=-2*x - y;q=x^2 + 9*x*y + 9*y^2 + 12*x - 6$
Possible approach is to equate coefficients symbolically.
