If $x+y=0$, then this equation becomes an identity. Otherwise, the given equation is equivalent to $$x+y = x^2-xy+y^2$$ which can be transformed to $$(x-y)^2 + (x-1)^2 + (y-1)^2=2$$ This can only hold when (i) two term equal $1$ and the other equals $0$, or (ii) one of them equals $2$ and the rest two equal $0$. Doing casework leads to five additional solutions $$(0, 1), (1,0), (1, 2), (2,1), (2,2)$$