For $n \geq 2$, I want to find all solutions $x_i$ of the equation

\begin{equation}
    \begin{cases}
        x_i \in [-1,1], 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 $n$?