As Fredholm determinants are more and more frequent in the context of stochastic integrability, I discuss in this talk the existence of a common framework in many integrable systems where they appear. This consists in a hierarchy of equations, akin to the Zakharov-Shabat system, connecting an integro-differential extension of the Painleve II hierarchy, the finite-time solutions of the KardarParisi-Zhang equation and multi-critical fermions at finite temperature. The talk is based on the results of the paper arXiv:2008.01509 Speaker(s): Alexandre Krajenbrink (SISSA)