Let $n$ be any integer divisible by and greater than 8. I have run into the question that whether there exist complex numbers $z_1,z_2,\ldots,z_{n-1}$ with modules $1$ satisfying the following quadratic equations, $\sum\limits_{r=1}^{2t-1}(-1)^rz_rz_{2t-r}+\sum\limits_{r=2t+1}^{n-1}(-1)^rz_rz_{n+2t-r}=\frac{2z_{2t}}{(n^2+n+1)^{\frac{1}{2}}},t=1,2,\ldots,\frac{n}{2}-1,$ $z_tz_{\frac{n}{2}+t}=z_{2t},t=1,2,\ldots,\frac{n}{2}-1,$ and $z_tz_{n-t}=(-1)^t,t=1,2,\ldots,\frac{n}{2}-1.$
What I seek for is a method which can deal with general $n$. So the Groebner basis theory seems hard to apply to this problem. Also, I wonder if translating the equations as well as the module constraints to quadratic equations of real variables by setting $z_j=x_j+iy_j$ is helpful.
