  Let T be a complete first order theory in a countable relational language L .
We assume relation symbols have been added to make each formula equivalent to a
predicate. Adjoin a new unary function symbol sigma to obtain the language
L_sigma; T_sigma is obtained by adding axioms asserting that sigma is an
L-automorphism. We provide necessary and sufficient conditions for T_sigma to
have a model companion when T is stable. Namely, we introduce a new condition:
T_sigma admits obstructions, and show that T_sigma has a model companion iff
and only if T_sigma does not admit obstructions. This condition is weakening of
the finite cover property: if a stable theory T has the finite cover property
then T_sigma admits obstructions.
