Let $x=r(\cos a+i\sin a),y=r(\cos b+i\sin b),z=r(\cos c+i\sin c)$ Given conditions lead to: $\sin a+\sin b+\sin c=0,\sin 3a+\sin 3b+\sin 3c=0$ Hence $\sin a\sin b\sin c=0$, either $\sin a,\sin b,\sin c=0$ If $\sin a=0,\sin b,\sin c\ne0$, then $\sin b=-\sin c$ Either $x,y,z$ are real or ($x$ is real and $y,z$ are conjugate pairs).