  We describe the group of $\mathbb Z$-linear automorphisms of the ring of
integers of a number field $K$ that preserve the set $V_{K,k}$ of $k$th
power-free integers: every such map is the composition of a field automorphism
and the multiplication by a unit.
  We show that those maps together with translations generate the extended
symmetry group of the shift space $\mathbb D_{K,k}$ associated to $V_{K,k}$.
Moreover, we show that no two such dynamical systems $\mathbb D_{K,k}$ and
$\mathbb D_{L,l}$ are topologically conjugate and no one is a factor system of
another.
  We generalize the concept of $k$th power-free integers to sieves and study
the resulting admissible shift spaces.
