  Recently, the separated fragment (SF) of first-order logic has been
introduced. Its defining principle is that universally and existentially
quantified variables may not occur together in atoms. SF properly generalizes
both the Bernays-Sch\"onfinkel-Ramsey (BSR) fragment and the relational monadic
fragment. In this paper the restrictions on variable occurrences in SF
sentences are relaxed such that universally and existentially quantified
variables may occur together in the same atom under certain conditions. Still,
satisfiability can be decided. This result is established in two ways: firstly,
by an effective equivalence-preserving translation into the BSR fragment, and,
secondly, by a model-theoretic argument.
  Slight modifications to the described concepts facilitate the definition of
other decidable classes of first-order sentences. The paper presents a second
fragment which is novel, has a decidable satisfiability problem, and properly
contains the Ackermann fragment and---once more---the relational monadic
fragment. The definition is again characterized by restrictions on the
occurrences of variables in atoms. More precisely, after certain
transformations, Skolemization yields only unary functions and constants, and
every atom contains at most one universally quantified variable. An effective
satisfiability-preserving translation into the monadic fragment is devised and
employed to prove decidability of the associated satisfiability problem.
