Let $n \geq 2$.
Could it be proved that the following system, with $z_k\in \mathbb C$,
$ \begin{cases} z_1^n + z_{n}z_1^{n-1} + z_{n-1}z_1^{n-2} + \cdots + z_2z_1+z_1 & = 0 \\ z_2^n + z_{n}z_2^{n-1} + z_{n-1}z_2^{n-2} + \cdots + z_2z_2+z_1 &= 0 \\ z_3^n + \color{blue}{z_{n}z_3^{n-1}} + z_{n-1}z_3^{n-2} + \cdots + z_2z_3+z_1 &=0 \\ \vdots &=\vdots\\ z_n^n + \color{blue}{z_{n}^{n}} + z_{n-1}z_n^{n-2} + \cdots +z_2z_n+z_1 &=0 \end{cases}$
has a finite set of solutions?
Remark: For $n=2$ it is easy, for $n=3$ a bit more work by doable.