This work deepens the study of integrable asymptotic symmetries for ${\mathrm{AdS}}_3$. They are given by an infinite set of integrable nonlinear equations known as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, characterized by an also infinite set of Abelian conserved charges. We present their field-dependent Killing vectors and the computation of the canonical charges associated to the asymptotic metric, together with their corresponding charge algebra. We study black hole thermodynamics and show that the temperature for stationary black holes falling in the AKNS asymptotics is always constant, even in the case where the solutions are not axisymmetric. The temperature is directly encoded in the spectral polynomial of the corresponding stationary Schrödinger problem, highlighting a precise connection between thermodynamic quantities and the spectral data of quantum integrable systems. This is related to the existence of a hyperelliptic curve, which appears as a fundamental object in many integrable systems. We also present a special solution associated with the Korteweg-de Vries (KdV) equation, that is a particular case of the AKNS integrable hierarchy. It is presented in the form of a periodic soliton leading to a cnoidal KdV black hole, whose temperature is characterized by two copies of hyperelliptic curves.

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• Classical black holes
• Gauge-gravity dualities
• Infinite-dimensional symmetries
• Integrable systems