  Axionic cosmic string solutions are investigated in a superstring motivated
model with a pseudo-anomalous U(1) gauge symmetry. The inclusion of a gauge
field and spatially varying dilaton allow local defect solutions with finite
energy per unit length to be found. Fermion zero modes (whose presence is
implied by supersymmetry) are also analysed. The corresponding fermion currents
suggest strong cosmological bounds on the model. It is shown that the unusual
form of the axion strings weakens these bounds. Other cosmological constraints
on the underlying theory are also discussed.
