# Is it possible that the following integral is $0$?

Given any integer $$n\geqslant1$$, let $$E,F$$ be two subsets of $$\{\{i,j\}:1\leqslant i such that every two sets in $$F$$ are disjoint.

It is not difficult to see that $$\int_{1<|z|<2}\prod_{\{i,j\}\in E}\left(\frac{z_i}{z_j}+\frac{z_j}{z_i}-2\right)dz\neq0$$ where $$|z|=\sqrt{|z_1|^2+\cdots+|z_n|^2}$$ and $$dz$$ denotes the Lebesgue measure on $$\mathbb{C}^n$$.

My question is whether it is possible that $$\int_{1<|z|<2}\prod_{\{i,j\}\in E}\left(\frac{z_i}{z_j}+\frac{z_j}{z_i}-2\right)\prod_{\{i,j\}\in F}|z_i-z_j|^2dz=0.$$ I would be very grateful for thoughts on how to tackle this problem. Any idea is welcome!