I'm interested in the complexity of a particular variant of 1-in-3 SAT. Assume, as is usual, that clauses are allowed to be of length 1, 2, or 3. Then add the restriction that for any clause of length 3, at least one variable must be "unique," meaning it must appear in no other clause.
Another way to phrase this is, for each clause, at most two variables can be "non-unique" and appear in other clauses.
What is the complexity of this variant of 1-in-3 SAT?