The number of solution $z^3+\overline{z}=0$  is

           $z^3+\overline{z}=0\Rightarrow z^3=-\overline{z}$

$\Rightarrow$       $|z{|}^3=|-\overline{z}|\Rightarrow |z{|}^3=\left|z\right|$

           $\left|z\right|\left(\right|z|-1)\left(\right|z|+1)=0$

$\Rightarrow$       $\left|z\right|=0$   or  $\left|z\right|=1$               $[\because |z|+1>0]$

If        $\left|z\right|=0,$  then $z=0$

If        $\left|z\right|=1$  we get  $|z{|}^2=1\Rightarrow z\overline{z}=1$

Thus, $z^3+\overline{z}=0\Rightarrow z^3+1/z=0$

$\Rightarrow$       $z^4+1=0$

This equation has four non-zero and distinct roots.

Therefore, the given equation has five roots.

TIP It is unnecessary to find roots of $z^4+1=0$