  The exchange operator formalism (EOF) describes many-body integrable systems
using phase-space variables involving an exchange operator that acts on any
pair of particles. We establish an equivalence between models described by EOF
and the complete infinite family of parent Hamiltonians (PHJ) describing
quantum many-body models with ground-states of Jastrow form. This makes it
possible to identify the invariants of motion for any model in the PHJ family
and establish its integrability, even in the presence of an external potential.
Using this construction we establish the integrability of the long-range
Lieb-Liniger model, describing bosons in a harmonic trap and subject to contact
and Coulomb interactions in one dimension. We give a variety of examples
exemplifying the integrability of Hamiltonians in this family.
