  Systems of equations with sets of integers as unknowns are considered. It is
shown that the class of sets representable by unique solutions of equations
using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in
T}$ and with ultimately periodic constants is exactly the class of
hyper-arithmetical sets. Equations using addition only can represent every
hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can
also be represented by equations over sets of natural numbers equipped with
union, addition and subtraction $S \dotminus T=\makeset{m-n}{m \in S, \: n \in
T, \: m \geqslant n}$. Testing whether a given system has a solution is
$\Sigma^1_1$-complete for each model. These results, in particular, settle the
expressive power of the most general types of language equations, as well as
equations over subsets of free groups.
