  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.
