  We prove that the simplest gravitational symmetry reduced models describing
cosmology and black holes mechanics are invariant under the Schr\"{o}dinger
group. We consider the flat FRW cosmology filled with a massless scalar field
and the Schwarzschild black hole mechanics, construct their conserved charges
using the Eisenhart-Duval (ED) lift method and show that they form a
Schr\"{o}dinger algebra. Our method illustrates how the ED lift and the more
standard approach analyzing the geometry of the field space are complementarity
in revealing different set of symmetries of these systems. We further identify
an infinite-dimensional symmetry for those two models, generated by conserved
charges organized in two copies of a Witt algebra. These extended charge
algebras provide a new algebraic characterization of these homogeneous
gravitational sectors. They guide the path to their quantization and open the
road to non-linear extensions of quantum cosmology and quantum black holes
models in terms of hydrodynamic equations in field space.
