# Map between curves and integral points

Suppose we have an equation of two variables $$x,y$$, for example $$E: x^3+y^3=d$$, and we have a bijective map \begin{align*} \varphi: E &\to F\\ (x,y) &\mapsto \left(\frac{3d}{x+y},-\frac{9dy}{x+y}\right) \end{align*} where $$F$$ is the (nonsingular) elliptic curve $$F : y^2=x^3-432d^2$$. If we prove that $$F$$ doesn't have integer solutions (or has finitely many) is that sufficient to prove that $$E$$ doesn't have solutions (or has finitely many solutions)? In other words, does the bijective map $$\varphi$$ from equation to elliptic curve preserve the structure between them?i don't mean this example i mean in general specialy Integer map .