Gauge transformations of an Eisenhart-Duval lift ($\mathbf{A}\mapsto \mathbf{A}+\nabla \chi ,u\mapsto u+2\chi$)

A time-independent gauge transformation $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ in an Eisenhart-Duval lift is the coordinate change $u\mapsto u+2\chi$. It preserves the metric one-form $du-2\mathbf{A}\cdot d\mathbf{x}-Udt$ and adds only $-2d\chi /dt$ to the reduced Lagrangian.