Solving complex numbers equation $z^3 = \overline{z} $ We have the following equation:
$$z^3 = \overline{z} $$
I set z to be $z = a + ib$ and since I know that $ \overline{z} = a - ib$. I was trying to solve it by opening the left side of the equation.
$$ z^3 = (a+ib)^3 \Rightarrow $$
$$ [a^2+b^2+i(ab + ba)](a+ib) \Rightarrow $$
$$ a^3 - b^2a - 2b^2a + i (2a^2b + b^2a - b^3) $$
but this is where I got so far and I'm not sure how continue and if my solution so far is even the right way to solve it.

one of solutions is obviusly $z=0$. For other solutions the simple way is to write $z=re^{ia}$, then
$$
z^3=\bar z\implies |z^3|=|\bar z|=|z|\implies |z|^2=1\implies r=1
$$
now we calculate $a$ by observing that
$$
z^4=z\bar z=1
$$
so you just need to find all $4$-th roots of unity to finish...
