For $n \geq 1$, I want to find all solutions $x_i$ of the equation

\begin{equation} \begin{cases} x_i \in \mathbb{Z}, i=0,1,2...,n-1 \\ x_i^2 = 1, i=0,1,2...,n-1 \\ \omega = \cos(2\pi/n)+i\sin(2\pi/n) \\ z = \sum_{i=0}^{n-1} x_i \omega^{i} \\ (z*\operatorname{Conj} z ) \in \mathbb{Z} \end{cases} \end{equation}

As an example, $x_i = 1, i=0,1,2...,n-1$ is one solution to this equation. And $x_i = -1, i=0,1,2...,n-1$ is another solution. For small $n$, all solutions can be found by mathematical software. Is there any good idea for bigger $n$?

Here is the computational result for small $n$:

Here is the further detail, the prime number related patterns are quite obvious.