Combinatorics of Schur ultrafilters

In this paper, we provide a combinatorial characterization of the elements of Schur ultrafilters on countable commutative groups. Using this characterization, we construct a free Schur ultrafilter on $\mathbb{Z}$ that is not infinitary Schur. Moreover, assuming the Continuum Hypothesis, we establish the existence of a free Schur P-point on $\mathbb{Z}$.