It is shown that the Ablowitz-Kaup-Newell-Segur (AKNS) integrable hierarchy can be obtained as the dynamical equations of three-dimensional general relativity with a negative cosmological constant. This geometrization of the AKNS system is possible through the construction of novel boundary conditions for the gravitational field. These are invariant under an asymptotic symmetry group characterized by an infinite set of AKNS commuting conserved charges. Gravitational configurations are studied by means of $SL(2,\mathbb{R})$ conjugacy classes. Conical singularities and black hole solutions are included in the boundary conditions.

中文翻译：

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可积系统和时空动力学

结果表明，Ablowitz-Kaup-Newell-Segur(AKNS)可积层级可以作为具有负宇宙常数的三维广义相对论的动力学方程得到。通过为引力场构建新的边界条件，AKNS 系统的这种几何化是可能的。这些在渐近对称群下是不变的，该群以无限集的 AKNS 交换守恒电荷为特征。引力配置的研究是通过$秒升(2,\mathbb{电阻})$共轭类。圆锥奇点和黑洞解包含在边界条件中。