Classical integrability in the presence of a cosmological constant: analytic and machine learning results

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. For a certain solution subspace, we demonstrate partial integrability by showing that a subset of the equations of motion in two dimensions are the compatibility conditions for a linear system. Subsequently, we study the integrability of these two-dimensional models from a complementary one-dimensional point of view, framed in terms of Liouville integrability. In this endeavour, we employ various machine learning techniques to systematise our search for numerical Lax pair matrices for these models, as well as conserved currents expressed as functions of phase space variables.