Classical Integrability in the Presence of a Cosmological Constant: Analytic and Machine Learning Results

Abstract

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