  We show that for every countable recursively saturated model $M$ of Peano
Arithmetic and every subset $A \subseteq M$, there exists a full satisfaction
class $S_A \subset M^2$ such that $A$ is definable in $(M,S_A)$ without
parametres. It follows that in every such model, there exists a full
satisfaction class which makes every element definable and thus the expanded
model is minimal and rigid. On the other hand, we show that for every full
satisfaction class $S$ there are two elements which have the same arithmetical
type, but exactly one of them is in $S$. In particular, the automorphism group
of a model expanded with a satisfaction class is never equal to the
automorphism group of the original model. The analogue of many of the results
proved here for full satisfaction classes were obtained by Roman Kossak for
partial inductive satisfaction classes. However, most of the proofs relied
heavily on the induction scheme in a crucial way, so recapturing the results in
the setting of full satisfaction classes requires quite different arguments.
