Input: System of linear equations $$A[x_1,\dots,x_{t}]=b$$ where number of equations is at least number of variables but independence is not guaranteed. However there is atmost one non-negative solution which is $0/1$ solution but there could be other mixed signs solutions. If we represent $Ax=b$ as a polyhedral inequality $\tilde Ax\leq\tilde b$:
The constraints $\tilde A,\tilde b$ are integer entry matrices.
Every row of $\tilde A$ is $0/-1$ or $0/1$ ($1$ occurs iff $-1$ does not occur) and number of non-zero entries is $2,4,6$ per row.
For $\tilde A$ two $4$-entry rows do not intersect, a $6$-entry row and a $4$-entry row intersect in $3$ columns, and $2$-entry rows intersect with other rows in $1$ or $2$ columns.
For 4-entry rows of $\tilde A$ having no $-1$ the corresponding entries of $\tilde b$ contain $2$, for $2$-entry rows of $\tilde A$ having no $-1$ the corresponding entries of $\tilde b$ contain $1$, and for $6$-entry rows of $\tilde A$ having no $-1$ the corresponding entries of $\tilde b$ contain $4$.
If row of $\tilde A$ is $-1$ replace the rules in 4. by entries of $\tilde b$ having negative $-2$ ,$-1$ and $-4$ correspondingly.
I could not identify a scenario where the system is not totally unimodular but I think it is not $TDI$ or totally unimodular.
Output: Either the unique non-negative solution or certify no solutions exist.
I. Is there a polynomial complexity solution for above problem?
II. If number of $0/1$ solutions is not $\leq1$ is it possible to count number of solutions mod $2$ in polynomial complexity of all non-negative solutions are $0/1$?