67th NMO Selection Tests for JBMO

Rewrite the given condition successively $(x+y{)}^3-3xy(x+y)=x^3{y}^3-3x^2{y}^2$, i.e. $(x+y{)}^3-(xy{)}^3=3xy(x+y)-3x^2{y}^2$, or $(x+y-xy)(x^2+2xy+y^2+x^{2}y+x{y}^2+x^2{y}^2)=3xy(x+y-xy)$. We either have $x+y=xy$, which leads to $E=1$ (obtained for $x=y=2$), or $x^2+2xy+y^2+x^{2}y+x{y}^2+x^2{y}^2=3xy$. The last equality, multiplied by $2$, can be written equivalently $x^2(y+1{)}^2+y^2(x+1{)}^2+(x-y{)}^2=0$, which is possible if and only if $x=y=-1$ ($x=y=0$ is forbidden). In this case, $E=-2$. In conclusion, the set of the possible values of $E$ is $\{-2,1\}$.