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 $n$:

| $n$ | Number of solutions | $2^n$ | Percentage |
|:----------:|:------:|:------:|:------:|
|$1$|$2$|$2$|$100.00\%$|
|$2$|$4$|$4$|$100.00\%$|
|$3$|$8$|$8$|$100.00\%$|
|$4$|$16$|$16$|$100.00\%$|
|$5$|$12$|$32$|$37.50\%$|
|$6$|$64$|$64$|$100.00\%$|
|$7$|$44$|$128$|$34.38\%$|
|$8$|$144$|$256$|$56.25\%$|
|$9$|$80$|$512$|$15.63\%$|
|$10$|$244$|$1024$|$23.83\%$|
|$11$|$68$|$2048$|$3.32\%$|
|$12$|$1816$|$4096$|$44.34\%$|
|$13$|$132$|$8192$|$1.61\%$|
|$14$|$2020$|$16384$|$12.33\%$|
|$15$|$1628$|$32768$|$4.97\%$|
|$16$|$4480$|$65536$|$6.84\%$|
|$17$|$36$|$131072$|$0.03\%$|
|$18$|$17200$|$262144$|$6.56\%$|
|$19$|$116$|$524288$|$0.02\%$|
|$20$|$33416$|$1048576$|$3.19\%$|
|$21$|$6644$|$2097152$|$0.32\%$|
|$22$|$30364$|$4194304$|$0.72\%$|
|$23$|$140$|$8388608$|$0.00\%$|
|$24$|$530512$|$16777216$|$3.16\%$|
|$25$|$1032$|$33554432$|$0.00\%$|
|$26$|$173164$|$67108864$|$0.26\%$|
|$27$|$14336$|$134217728$|$0.01\%$|