An example: Let $x_1,x_2,x_3$ be variables satisfying the equations given by \begin{align*} \begin{bmatrix} x_1&x_3&x_2\\ x_3&x_2&x_1\\ x_2&x_1&x_3 \end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix}=\begin{bmatrix} x_1\\x_2\\x_3\end{bmatrix} \end{align*} and we want to find solutions in $\mathbb{Z}$ satisfying the equations gotten from equating the entries and $x_1+x_2+x_3=1$.
Then we can show that this has only solutions $(1,0,0),(0,1,0),(0,0,1)$. This can be obtained from the equation $x_2^2-x_3^2=x_2-x_3$.
Question: Now suppose we want to find solution of the equations
\begin{align*} \begin{bmatrix} x_1&x_4&x_2&x_5&x_3\\ x_4&x_2&x_5&x_3&x_1\\ x_2&x_5&x_3&x_1&x_4\\ x_5&x_3&x_1&x_4&x_2\\ x_3&x_1&x_4&x_2&x_5 \end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\\x_4\\x_5 \end{bmatrix}=\begin{bmatrix} x_1\\x_2\\x_3\\x_4\\x_5\end{bmatrix} \end{align*} and we want to find solutions in $\mathbb{Z}$ satisfying the equations gotten from equating the entries and $x_1+x_2+x_3+x_4+x_5=1$.
How to solve this system of equation?
Substituting $x_5=1-(x_1+x_1+x_3+x_4)$ we obtain five polynomial equations in the variables $x_1,x_2,x_3,x_4$. All solutions over the complex numbers are given as follows: $$ (x_1,\ldots ,x_5)=(1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,0,0,0,1),(1/5,1/5,1/5,1/5,1/5). $$ Here the only non integer solution is $x_i=\frac{1}{5}$ for all $i$. This follows easily from using a Groebner basis.