The parameterized satisfiability problem over a set of Boolean

relations Gamma (SAT(Gamma)) is the problem of determining

whether a conjunctive formula over Gamma has at least one

model. Due to Schaefer's dichotomy theorem the computational

complexity of SAT(Gamma), modulo polynomial-time reductions, has

been completely determined: SAT(Gamma) is always either tractable

or NP-complete. More recently, the problem of studying the

relationship between the complexity of the NP-complete cases of

SAT(Gamma) with restricted notions of reductions has attracted

attention. For example, Impagliazzo et al. studied the complexity of

k-SAT and proved that the worst-case time complexity increases

infinitely often for larger values of k, unless 3-SAT is solvable in

subexponential time. In a similar line of research Jonsson et al.

studied the complexity of SAT(Gamma) with algebraic tools borrowed

from clone theory and proved that there exists an NP-complete problem

SAT(R^{neq,neq,neq,01}_{1/3}) such that there cannot exist any NP-complete SAT(Gamma) problem with strictly lower worst-case time complexity: the easiest NP-complete SAT(Gamma) problem. In this paper

we are interested in classifying the NP-complete SAT(Gamma)

problems whose worst-case time complexity is lower than 1-in-3-SAT but

higher than the easiest problem SAT(R^{neq,neq,neq,01}_{1/3}). Recently it was conjectured that there only exists three satisfiability problems of this form. We prove that this conjecture does not hold and that there is an infinite number of such SAT(Gamma) problems. In the process we determine several algebraic properties of 1-in-3-SAT and related problems, which could be of independent interest for constructing exponential-time algorithms.