  We consider fermions defined on a continuous one-dimensional interval and
subject to weak repulsive two-body interactions. We show that it is possible to
perturbatively construct an extensive number of mutually compatible conserved
charges for any interaction potential. However, the contributions to the
densities of these charges at second order and higher are generally non-local
and become spatially localized only if the potential fulfils certain
compatibility conditions. We prove that the only solutions to the first of
these conditions are the Cheon-Shigehara potential (fermionic dual to the
Lieb-Liniger model) and the Calogero-Sutherland potentials. We use our
construction to show how Generalized Hydrodynamics (GHD) emerges from the
Bogoliubov--Born--Green--Kirkwood--Yvon hierarchy, and argue that GHD in the
weak interaction regime is robust under non-integrable perturbations.
