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

[anemone]

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

$k^2=x+y$ and $k^3=x^2+y^2$

 

[anemone]

Hint:

Spoiler

Compare $2(x^2+y^2)$ and $(x+y)^2$.