Is 1-in-3 SAT NP-complete if each literal must appear the same number of times as its negation? That is, $x_i$ must appear the same number of times as $\neg x_i$ for all $i$.
1-in-3 SAT is a decision problem in which a set of clauses must all be satisfied. Each clause contains three literals, and in order to be satisfied, a clause must contain two false literals and one true literal.
1-in-3 SAT is NP complete. 3 SAT is also NP-complete even when each literal is required to appear the same number of times as its negation. Is 1-in-3 SAT also NP-complete with this restriction? I've seen reductions for 1-in-3 SAT and for variable balancing, but I'm having trouble putting them together. Any help would be appreciated.
Thank you.