MHB Solve System of n Equations: Can You Help?

[Mathick]

Solve the system of $$n$$ equations with unknown $$x_1, x_2, ... , x_n$$, for $$n\ge 2$$:

$$2x_1^3 + 4 =x_1^2 (x_2 +3)$$
$$2x_2^3 + 4 =x_2^2 (x_3 +3)$$
$$......$$
$$2x$$_n-1³$$+ 4 = $$$$x$$_n-1²$$(x_n +3)$$
$$2x_n^3 + 4 =x_n^2 (x_1 +3)$$

Can you help me?

 

[Ackbach]

The symmetry of the equations led me to wonder what would happen if all the variables were equal. In that case, there are two solutions. $x_j=2$ is a double-root, and $x_j=-1$ is a single root. I don't think there is any guarantee that these are the only solutions, but you can see by inspection that they both solve the system.