The main goal of this work is to pursue an investigation of cosmic string configurations focusing on possible consequences of the Lorentz-symmetry breaking by a background vector. We analyze the possibility of cosmic strings as a viable source for fermionic Cold Dark Matter particles. Whenever the latter are charged and have mass of the order of $10^{13}GeV$, we propose they could decay into usual cosmic rays. We have also contemplated the sector of neutral particles generated in our model. Indeed, being neutral, these particles are hard to be detected; however, by virtue of the Lorentz-symmetry breaking background vector, it is possible that they may present an electromagnetic interaction with a significant magnetic moment.
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