We show that the one-point distribution functions of some classical integrable KPZ models, including the Totally Asymmetric Simple Exclusion Process and Reflected Brownian Motions/Brownian Last Passage Percolation, satisfy a new family of nonlinear evolution equations. Moreover, these evolution equations may be regarded as reparametrizations or as scaling limits of the Hirota Bilinear Difference Equation, a known universal discrete master equation in integrable systems theory from which many other key integrable equations, including the Kadomtsev-Petviashvili (KP) equation and the two-dimensional Toda Lattice equation (2DTL) previously linked to the KPZ universality class, can themselves be derived. This new explicit connection allows us to import tools from classical integrability theory, and as an application we derive zero-curvature conditions/Lax pairs as well as perform formal asymptotics on the evolution equations. On the other hand, our work develops a general solution framework to the equations, providing a class of solutions to the HBDE not previously studied in classical integrable systems theory. This suggests the distribution functions of integrable KPZ models may represent a hierarchical system of equations running parallel to the deeply influential hierarchies in classical soliton theory.