While all first-order theories have plenty of models with many automorphisms (e.g., saturated), models with few automorphisms are harder to come by, and their existence varies with the theory. In the extreme case of rigid models (= with no nontrivial automorphism), some theories have no rigid models at all (such as divisible ordered abelian groups), while e.g. Peano arithmetic has many: every model of PA has a rigid elementary end-extension of the same cardinality.
In this talk, we will give a complete description of rigid models of Presburger arithmetic Th(Z,+,<), and its toy version Th(Z,+). As we will see, Presburger arithmetic has rigid models that are somewhat nontrivial, but it only gets so far; in particular, it has no rigid models larger than the continuum.