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Pole-skipping is a property of gravitational waves dictated by theirbehaviour at horizons of black holes. It stems from the inability tounambiguously impose ingoing boundary conditions at the horizon at an infinitediscrete set of Fourier modes. The phenomenon has been best understood, whensuch a description exists, in terms of dual holographic (AdS/CFT) correlationfunction that take the value of `$0/0$' at these special points. In this work,we investigate details of pole-skipping purely from the point of view ofclassical gravity in 4$d$ massive black hole geometries with flat, sphericaland hyperbolic horizons, and with an arbitrary cosmological constant. We showthat pole-skipping points naturally fall into two categories: the algebraicallyspecial points and a set of pole-skipping points that is common to the even andodd channels of perturbations. Our analysis utilises and generalises (toarbitrary maximally symmetric horizon topology and cosmological constant) the`integrable' structure of the Darboux transformations, which relate the masterfield equations that describe the evolution of gravitational perturbations inthe two channels. Finally, we provide new insights into a number of specialcases: spherical black holes, asymptotically Anti-de Sitter black branes andpole-skipping at the cosmological horizon in de Sitter space.

Pole-skipping of gravitational waves in the backgrounds of four-dimensional massive black holes