<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] Write $\Delta\geq0$. 
It must help!
$$x^2-(y+1)x+y^2-y=0,$$
which gives $$(y+1)^2-4(y^2-y)\geq0$$ or
$$3y^2-6y-1\leq0$$ or
$$1-\frac{2}{\sqrt3}\leq y\leq1+\frac{2}{\sqrt3},$$
which gives $$0\leq y\leq2,$$
which gives all solutions:
For $y=0$ we get $x^2-x=0$, which gives $(0,0)$ and $(1,0)$.
For $y=1$ we get $x^2-2x=0$, which gives $(0,1)$ and $(2,1)$.
For $y=2$ we get $x^2-3x+2=0$, which gives $(2,2)$ and $(1,2).$
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