We study the integrability of two-dimensional theories that are obtained by a
dimensional reduction of certain four-dimensional gravitational theories
describing the coupling of Maxwell fields and neutral scalar fields to gravity
in the presence of a potential for the neutral scalar fields. By focusing on a
certain solution subspace, we show that a subset of the equations of motion in
two dimensions are the compatibility conditions for a modified version of the
Breitenlohner-Maison linear system. Subsequently, we study the Liouville
integrability of the 2D models encoding the chosen 4D solution subspace from a
one-dimensional point of view by constructing Lax pair matrices. In this
endeavour, we successfully employ a linear neural network to search for Lax
pair matrices for these models, thereby illustrating how machine learning
approaches can be effectively implemented to augment the identification of
integrable structures in classical systems.

通过降维，我们研究了由某些四维引力理论的降维而来的二维理论的可积性，重点研究了描述最大韦尔斯场和中性标量场与引力在中性标量场势存在的情况下的耦合的四维引力理论的解决子空间，并通过构造 Lax 对矩阵从一维角度研究了选择的四维解决子空间的二维模型的 Liouville 可积性，结果表明机器学习方法可以有效地用于增强经典系统中可积结构的识别。