  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 Painlev\'e property of the
latter would be implemented in terms of the so-called Painlev\'e-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 Painlev\'e-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.
