  Spin Matrix theory (SMT) limits provide a way to capture the dynamics of the
AdS/CFT correspondence near BPS bounds. On the string theory side, these limits
result in non-relativistic sigma models that can be interpreted as novel
non-relativistic strings. This SMT string theory couples to non-relativistic
$U(1)$-Galilean background geometries. In this paper, we explore the relation
between pp-wave backgrounds obtained from Penrose limits of AdS${}_5 \times
S^5$, and a new type of $U(1)$-Galilean backgrounds that we call flat-fluxed
(FF) backgrounds. These FF backgrounds are the simplest possible SMT string
backgrounds and correspond to free magnons from the spin chain perspective. We
provide a catalogue of the $U(1)$-Galilean backgrounds one obtains from SMT
limits of string theory on AdS${}_5 \times S^5$ and subsequently study large
charge limits of these geometries from which the FF backgrounds emerge. We show
that these limits are analogous to Penrose limits of AdS${}_5 \times S^5$ and
demonstrate that the large charge/Penrose limits commute with the SMT limits.
Finally, we point out that $U(1)$-Galilean backgrounds prescribe a symplectic
manifold for the transverse SMT string embedding fields. This is illustrated
with a Hamiltonian derivation for the SMT limit of a particle.
