Given $\vec{a}, \vec{b} \in \mathbb{Z}^d$ and a system of two-equation: $$ \begin{cases} \langle \vec{a}, \vec{x} \rangle = 0 \\ \langle \vec{b}, 1-\vec{x} \rangle = 0 \end{cases} $$ where $\vec{x} \in \{0, 1\}^d$. Here $\langle,\rangle$ refers to inner product and $1-\vec{x}$ referrs to all 1 vector minus $\vec{x}$.
How can I verify such a system exists a solution for $\vec{x}$?
If there are solutions, how can I enumerate all of them?
For small $d$ you can of course enumerate all $x\in \lbrace 0,1 \rbrace^d$ and try them all in the system of equations.
More generally, to determine if a solution exists you can treat the problem as a binary integer program with a trivial objective function (minimize 0 subject to your two equations) and solve it with any MIP solver. One (time-consuming) way to enumerate all solutions is to solve the IP model, add a constraint prohibiting the solution you got (and only that solution), and repeat. To eliminate a solution $x=\hat{x}$, you add the constraint $$\sum_{i:\hat{x}_i=0} x_i + \sum_{j:\hat{x}_j=1} (1-x_j) \ge 1.$$