Integrability of Conformal Killing Vectors in the Eisenhart Lift of Scalar-Field FLRW Cosmology

Abstract

We study the integrability conditions of the conformal Killing equations for the Eisenhart lift of a scalar field in a flat Friedmann-Lema\^\i tre-Robertson-Walker universe. We show that the potential found in our earlier work is already the most general local potential that admits a non-trivial conformal Killing vector in the sector independent of the cyclic Eisenhart coordinate. The determinant condition of the prolonged conformal Killing equations reduces to a nonlinear second-order differential equation for $h=V'/V$. We solve this equation locally and find two branches. The regular branch reproduces exactly the family of potentials obtained before, while the singular branch lies on the locus where the determinant equation cannot be written locally in normal form with respect to $h''$ and is incompatible with the full conformal Killing equations. Hence the ansatz used in our earlier work is exhaustive.