MHB Solving for $k$: $k^2=x+y$ and $k^3=x^2+y^2$

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Find all non-negative integers $k$ such that there are integers $x$ and $y$ with the property

$k^2=x+y$ and $k^3=x^2+y^2$
 
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Hint:

Compare $2(x^2+y^2)$ and $(x+y)^2$.
 

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