The waveform of a binary black hole coalescence appears to be both simple and universal. In this essay we argue that the dynamics should admit a separation into “fast and slow” degrees of freedom, such that the latter are described by an integrable system of equations, accounting for the simplicity and universality of the waveform. Given that Painlevé transcendents are a smoking gun of integrable structures, we propose the Painlevé-II transcendent as the key structural element threading a hierarchy of asymptotic models aiming at capturing different (effective) layers in the dynamics. Ward’s conjecture relating integrable and (anti-)self-dual solutions can provide the avenue to encode background binary black hole data in (nonlocal) twistor structures.