Integer Parities Generalization

If $x_1, x_2, \dots, x_n$ are integers, and the system of linear equations consists of all $\binom{n}{k}$ possible sums of $k$ variables, where:

• $n \geq 3$ and $1 < k < n$
• The coefficients are all 1
• Each of the sums equals an integer

Which of the following must be true about the system of equations?