  We construct the basic Neveu-Schwarz (NS) brane solutions of non-relativistic
string theory using longitudinal T-duality as a solution generating technique.
Extending the NS background fields to a supergravity multiplet, we verify that
all solutions we find are half-supersymmetric. The two perturbative solutions
we find both have an interpretation as the background geometry outside a
string-like object. Correspondingly, we refer to these non-Lorentzian
backgrounds as winding string and unwound string solution. Whereas the winding
string is part of the on-shell spectrum of non-relativistic string theory, the
unwound string only makes sense off-shell where it mediates the instantaneous
gravitational force. Seen from the nine-dimensional point of view, we find that
the winding string solution is sourced by a non-relativistic massive particle
and that the unwound string solution is sourced by a massless Galilean particle
of zero colour and spin. We explain how these two string solutions fit into a
discrete lightcone quantization of string theory. We shortly discuss the basic
NS five-brane and Kaluza-Klein monopole solutions and show that they are both
half-supersymmetric.
