Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. We say that an integer $n$ is primitively represented by $F$ if there exist coprime integers $x$ and $y$ such that $F(x,y)=n$. We also say that a subset $\mathcal S \subset \mathbb Z$ is primitively represented by $F$ if every $n \in \mathcal S$ is primitively represented by $F$.

Next, for any matrix

$$A = \left(\begin{matrix}a & b\\c & d\end{matrix}\right)$$

with rational entries, define

$$F_A(x,y) = F(ax + by, cx + dy).$$

We say that two binary forms $F$ and $G$ are _equivalent_ if there exists $A \in \operatorname{GL}_2(\mathbb Z)$ such that $F_A = G$.

I am interested in the following question: _given two distinct non-equivalent irreducible binary forms $F$ and $G$ of the same degree $d \geq 3$ and discriminant, when is it possible for them to represent primitively the same infinite subset of the integers?_

It has been pointed out to me by @StanleyYaoXiao that from a given binary form $F$ it is possible to construct $G$ satisfying the above criteria whenever $F$ has a non-trivial $\operatorname{GL}_2(\mathbb Q)$-automorphism (i.e. $A \in \operatorname{GL}_2(\mathbb Q) \setminus \operatorname{GL}_2(\mathbb Z)$ such that $F_A = F$). After our discussion, I suggested that the following special case, which I am particularly interested in, might be true:

_Let $T, S \in \operatorname{M}_2(\mathbb Z)$, $|\det S|=|\det T| > 1$, $T \neq S U$ for any $U \in \operatorname{GL}_2(\mathbb Z)$. Then $F_T$ and $F_S$ primitively represent the same infinite subset of the integers if and only if $F$ has no non-trivial $\operatorname{GL}_2(\mathbb Q)$-automorphism._

At the moment, I don't know how to prove or disprove this statement. It might be the case that some additional conditions on $F$, $T$ or $S$ should be introduced to make it correct.

P.S. A possible approach to answer this question would be to classify the rational points on the variety of the form

$$
F(x,y) - F(x',y')=0
$$

in $\mathbb P^3$. Perhaps, you can recommend me a source where a similar problem was considered?