  Nonlocal operators that have appeared in a variety of physical models satisfy
identities and enjoy a range of properties similar to their classical
counterparts. In this paper we obtain Helmholtz-Hodge type decompositions for
two-point vector fields in three components that have zero nonlocal curls, zero
nonlocal divergence, and a third component which is (nonlocally) curl-free and
divergence-free. The results obtained incorporate different nonlocal boundary
conditions, thus being applicable in a variety of settings.
