Functional equation $f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$ Find all of the functions defined on the set of integers and receiving the integers value, satisfying the condition:
$$f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$$
for each pair of integers $(a,b)$.

An easier answer.
Using $x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$ and putting $x=f(a+b),y=-f(a),z=-f(b)$ we get that $x^3+y^3+z^3-3xyz=0$ so we have that either $x+y+z=0$ or $x^2+y^2+z^2-xy-yz-zx=0$. If $x+y+z=0$ then we have that $f(a+b)=f(a)+f(b)$ which gives us $f(a)=af(1)$. If the second one is true we get from it that $x=y=z$, which means that $-f(a+b)=f(a)=f(b)$. From which we can get that $f(x)=0$ fot evey real number x. So those are the two solutions.
