Schaefer (1978) introduced a generalized satisfiability problem SAT (S) and showed that, depending on the nature of relations in S, SAT (S) is either in P or NP-complete. A similar result holds for generalized satisfiability with constants SAT_ {C}(S)(the version of the above problem where constants are allowed). We study the possibility to obtain a version of Schaefer's dichotomy theorem for instances satisfying an additional constraint, namely each variable appears at most twice. We prove several partial results on the complexity of the versions SAT (S, 2), SAT_ {C}(S, 2) of the above two problems that take into account this restriction. We obtain a dichotomy theorem for SAT_ {C}(S, 2) in the case when all relations in S are symmetric.