  The relation between the separability of a system of charged particles in a
uniform magnetic field and Galilean symmetry is revisited using Duval's
"Bargmann framework". If the charge-to-mass ratios of the particles are
identical, $e_a/m_a=\epsilon$ for all particles, then the Bargmann space of the
magnetic system is isometric to that of an anisotropic harmonic oscillator.
Assuming that the particles interact through a potential which only depends on
their relative distances, the system splits into one representing the center of
mass plus a decoupled internal part, and can be mapped further into an isolated
system using Niederer's transformation. Conversely, the manifest Galilean boost
symmetry of the isolated system can be "imported" to the oscillator and to the
magnetic systems, respectively, to yield the symmetry used by Gibbons and Pope
to prove the separability. For vanishing interaction potential the isolated
system is free and our procedure endows all our systems with a hidden
Schroedinger symmetry, augmented with independent internal rotations. All these
properties follow from the cohomological structure of the Galilei group, as
explained by Souriau's "d\'ecomposition barycentrique"
