The binary black hole merger waveform is both simple and universal. Adopting an effective asymptotic description of the dynamics, we aim at accounting for such universality in terms of underlying (effective) integrable structures. More specifically, under a ``wave-mean flow'' perspective, we propose that fast degrees of freedom corresponding to the observed waveform would be subject to effective linear dynamics, propagating on a slowly evolving background subject to (effective) non-linear integrable dynamics. The Painlevé property of the latter would be implemented in terms of the so-called Painlevé-II transcendent, providing a structural link between i) orbital (in particular, EMRI) dynamics in the inspiral phase, ii) self-similar solutions of non-linear dispersive Korteweg-de Vries-like equations (namely, the `modified Korteweg-de Vries' equation) through the merger and iii) the matching with the isospectral features of black hole quasi-normal modes in late ringdown dynamics. Moreover, the Painlevé-II equation provides also a `non-linear turning point' problem, extending the linear discussion in the recently introduced Airy approach to binary black hole merger waveforms. Under the proposed integrability perspective, the simplicity and universality of the binary black hole merger waveform would be accounted to by the `hidden symmetries' of the underlying integrable (effective) dynamics. In the spirit of asymptotic reasoning, and considering Ward's conjecture linking integrability and self-dual Yang-Mills structures, it is tantalizing to question if such universal patterns would reflect the actual full integrability of a (self-dual) sector of general relativity, ultimately responsible for the binary black hole waveform patterns.