  We introduce a class of J-like compacts of the plane with the property that
either any non-trivial local holomorphic family of symmetries of such compact
is finite or the compact is locally "laminar". The proof repeats the one for
the Julia set. Apart of Julia sets, this class includes attractors of
holomorphic iterated function systems with the strong separation condition. We
apply this to study the set of such iterated function systems having the same
attractor
