A cubic diophantine Equation

Question

While reading Diophantine equations I came across the following equation $$x^3+cy^3-3yx=0$$ Is there any known method to solve this equation for any $c$?

Answer 1

We have $y\mid x^3$ and $x\mid cy^3$. If we assume $y=ax$ for some integer $a$ then substituting this in $x^3+cy^3=3xy$ yields
$$ x=a(3-ac), \quad y=a^2(3-ac). $$ Conversely this is a solution of $x^3+cy^3=3xy$ for all integers $a$. This is a first step to solve this Diophantine equation (which is not homogeneous, i.e., not given by a binary cubic).