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As Fredholm determinants are more and more frequent in the context ofstochastic integrability, we unveil the existence of a common framework in manyintegrable systems where they appear. This consists in a quasi-universalhierarchy of equations, partly unifying an integro-differential generalizationof the Painlev\'e II hierarchy, the finite-time solutions of theKardar-Parisi-Zhang equation, multi-critical fermions at finite temperature anda notable solution to the Zakharov-Shabat system associated to the largest realeigenvalue in the real Ginibre ensemble. As a byproduct, we obtain the explicitunique solution to the inverse scattering transform of the Zakharov-Shabatsystem in terms of a Fredholm determinant.

From Painlev\'e to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies