  The Toda chain of nearest neighbour interacting particles on a line can be
described both in terms of geodesic motion on a manifold with one extra
dimension, the Eisenhart lift, or in terms of geodesic motion in a symmetric
space with several extra dimensions. We examine the relationship between these
two realisations and discover that the symmetric space is a generalised,
multi-particle Eisenhart lift of the original problem, that reduces to the
standard Eisenhart lift. Such generalised Eisenhart lift acts as an inverse
Kaluza-Klein reduction, promoting coupling constants to momenta in higher
dimension. In particular, isometries of the generalised lift metric correspond
to energy preserving transformations that mix coordinates and coupling
constants. A by-product of the analysis is that the lift of the Toda Lax pair
can be used to construct higher rank Killing tensors for both the standard and
generalised lift metrics.
