<s>[INST] Finding integral solutions of $x+y=x^2-xy+y^2$ 
Find integral solutions of $$x+y=x^2-xy+y^2$$

I simplified the equation down to 
$$(x+y)^2 = x^3 + y^3$$
And hence found out solutions $(0,1), (1,0), (1,2), (2,1), (2,2)$ but I dont think my approach is correct . Is further simplification required? Is there any other method to solve this? I am thankful to those who answer!
 [/INST] $x+y=x^2-xy+y^2
$.
Since this is
symmetrical in
$x$ and $y$,
we can assume that
$x \le y$.
If $x=y$,
this becomes
$2x = x^2$,
so
$x=0$ or $x=2$.
If $x < y$,
then
$2y
\gt x+y
=(x-y/2)^2+3y^2/4
\ge 3y^2/4
$.
There is no solution if
$y \le 0$.
If $ > 0$,
$2 \gt 3y/4
$
or
$y \lt 8/3$
so
$y \le 2$.
If $y=2$,
then
$x=0$ or $x=1$;
of there,
only $x=1$ works.
If $y=1$,
then $x=0$ works.
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