This might be related to an open problem.
Let $F(x,y)$ be homogeneous degree 4 squarefree polynomial with integer coefficients and $h(x,y)=a_1 x + a_2 y$ and $\gcd(F,h)=1$ and $k$ be integer.
Consider the projective curve over the rationals: $$ C : F(x,y)-k h(x,y)^2 z^2=0$$
$C$ is genus one curve and it might have infinitely many solutions $(x,y,z)$ coming from the group law.
Q1 Are there choices of $F,h,k$ such that $C$ infinitely many integer solutions $(x,y)$ with $x,y$ coprime (or possibly with $\gcd(x,y)$ bounded)?
Probably this is feasible with brute force approach and a single solution will be of interest to us.
We suspect solution exists.
