  As Fredholm determinants are more and more frequent in the context of
stochastic integrability, we unveil the existence of a common framework in many
integrable systems where they appear. This consists in a quasi-universal
hierarchy of equations, partly unifying an integro-differential generalization
of the Painlev\'e II hierarchy, the finite-time solutions of the
Kardar-Parisi-Zhang equation, multi-critical fermions at finite temperature and
a notable solution to the Zakharov-Shabat system associated to the largest real
eigenvalue in the real Ginibre ensemble. As a byproduct, we obtain the explicit
unique solution to the inverse scattering transform of the Zakharov-Shabat
system in terms of a Fredholm determinant.
