Positive 1-in-3 SAT FPT or Fixed Parameter Intractable
No, since even the generalization to 1-in-at-most-3 is FPT for the same reason as vertex cover.

(With at most 3 variables per clause, there will be at most 3 cases for each step.)

I feel like that generalization should have a simple

poly-size kernel, but can't figure out any way of showing that.

The corresponding problem for at-most-1-in-3 constraints, where one wants

exactly k variables to be true, is W[1]-hard by reduction from independent set:

If k&lt;2 then brute force, else:

There's a variable for each vertex and one other variable.

The constraints are ​ u , other_variable , v ​ for edges u,v.

Since 2≤k and other_variable is in every constraint, other_variable must be false, so

the satisfying assignments with exactly k Trues correspond to the independent k-sets.