@TedShifrin quick question about part iii). I know already that it is symmetric through $y=x$, I also have derived the reflection map, based on all of our previous work and redoing it here as exercise. To verify, is it enough to just use the coordinates given by my reflection and put them into the original equation to see that it works?

That is if $R(x,y) = (y,x)$ is the reflection map $R(t_{1}, t_{2}) = (t_{2}, t_{1})$ and then $t_{2}^{3} + t_{1}^{3} = 3t_{2}t_{1}$. Where $t_{1}, t_{2}$ is the parameterization of the expression. I'm just not writing them out explicitly but I did get them.