$\quad x^3 + y^3 + z^3 = 3xyz$
$\to x^3 + y^3 + z^3 - 3xyz=0$
$\to (x+y)^3 - 3xy(x+y)+ z^3 - 3xyz=0$
$\to (x+y+z)[(x+y)^2 - z(x+y) + z^2] - 3xy(x+y+z)=0$
$\to (x+y+z)(x^2 + 2xy + y^2 - zx - yz + z^2 - 3xy)=0$
$\to (x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx)=0$
$\to (x+y+z)(2x^2 + 2y^2 + 2z^2 - 2xy - 2yz - 2zx)=0$
$\to (x+y+z)[(x^2 - 2xy + y^2) + (y^2 - 2yz + z^2) + (z^2 - 2zx + x^2)]=0$
$\to (x+y+z)[(x-y)^2 + (y-z)^2 + (z-x)^2]=0$
$\to \left[\begin{array}{l}x + y + z = 0\\x = y = z\end{array}\right.$
