Let $I=\{i_0\} \cup J$ and $g(x)=1+x+x^2+x^4+\dots$.

We show that adding the relation $\sum_{i\in I}x_i=0$, i.e. $x_{i_0}=\sum_{i\ne i_0}x_i$ leads to an even degree power series: 

$$ f=g(x_{i_0})\prod_{j\in J} g(x_j) =\left(1+\sum_{i\in J} (g(x_i)-1)\right)\prod_{j\in J} g(x_j)\\ = \prod_{j\in J} g(x_j) + \sum_{i\in J}\left(g(x_i)-1\right)g(x_i)\prod_{j\in J\setminus \{i\}}g(x_j)\\ =  \prod_{j\in J} g(x_j) + \sum_{i\in J}x_i\prod_{j\in J\setminus \{i\}}g(x_j)= \prod_{j\in J} g(x_j) + \sum_{i\in J}x_i\frac{\partial}{\partial x_i}\prod_{j\in J}g(x_j)$$
Now this is an even degree power series: check that each monomial with an odd number of $x_j^1$ cancels.