Solve $ x^2 + xy + y^2 = \left( \frac{x+y}{3}+1 \right)^2 $ in integers

Question

Solve in integers $x,y \in \mathbb{Z}$ the equation: $$ x^2 + xy + y^2 = \left( \frac{x+y}{3}+1 \right)^2 $$ Could we solve this using lattice reduction?

Answer 1

Hint: Simplifying your given equation we get $$8x^2+7xy+8y^2-6x-6y-9=0$$ solving this equation for $y$ and considering the discriminat we get $$-23x^2+12x+36\geq 0$$ this is only fulfilled for $$x=-1,x=0,x=1$$