Consider $G:=\operatorname{GL}_3(\Bbb C)$ acting from the right on $R=\Bbb C[x_0,x_1,x_2]_3$ by linear substitution, i.e. we let $x=(x_0,x_1,x_2)^T$ and the action of $A\in G$ on $f\in R$ yields the polynomial $f(Ax)$. For polynomials $f,g\in R$ we write $f\sim g$ if there is a matrix $A\in G$ such that $f(Ax)=g(x)$. It is known that for every $f\in R$, there is an $a\in\Bbb C$ such that $f \sim f_a$, where $$ f_a := x_0^3 + x_1^3 + x_2^3 + a\cdot x_0x_1x_2. $$ Unfortunately for the non-German speakers, my only reference for this is the book Geometrische Methoden in der Invariantentheorie by Hanspeter Kraft, but I am reasonably sure this can be found elsewhere.
What I can not find is a proof for the following statement:
There are only finitely many pairs $(a,b)\in\Bbb C^2$ with $a\ne b$ and $f_a\sim f_b$.
For example, if there is some third root of unity $\zeta\in\Bbb C$ with $b=\zeta a$, then $f_b=f_a(\zeta x_0, x_1, x_2)$ but not much more can happen.
I am quite sure the statement is true, but I would be very interested to see a (direct/elementary) proof because I would like to prove something very similar and I am stuck.
Amendment. I am aware that one way to prove this is by showing that some polynomial in $a$ is a $G$-invariant function on $R$. I am looking for something even more elementary, because finding an invariant in my case seems hard. I am aware that the question is a bit fuzzy, and I deleted it because of that flaw - I undeleted it now because a fellow user asked me to, in private communication.