Given a 3SAT problem where each clause is a Monotone Clause (i.e. each clause with all positive or negative literals).
Moreover, we are promised that each solution for the above SAT problem (if any exists at all) will be of the form 1 in 3 SAT. In other words all valid solutions are promised to be Only Monotone Solutions.
Monotone 1 in 3 SAT is NP Complete, but with the additional promise above, does the problem still remain NP Complete (it obviously is NP) ?