  In this paper, we obtain two different static black string solutions by
considering as sources axisymmetric dark matter distributions in 3+1
dimensions. These solutions tend asymptotically to the usual static and
uncharged black string vacuum solution predicted by General Relativity (GR). We
show that both the solutions present an event horizon each, like the vacuum
solution, which is larger than the horizon of the latter. Then, we obtain the
Hawking temperature associated with the black string solutions. Differently
from what occurs with the static black string in the vacuum, we find that there
exists a linear density of mass (or tension) remnant associated with a
vanishing Hawking temperature for the obtained solutions. Thus, we analyze how
the presence of dark matter affects the occurrence of the remnants. Further, we
calculate other thermodynamic quantities, namely entropy, heat capacity, and
free energy per length unit, showing that thermal phase transitions can occur
in the presence of dark matter. We also analyze the weak (and null) energy
conditions and conclude that the dark matter does not behave like an exotic
fluid. Finally, we obtain the corresponding stationary solutions, determining
their tensions as functions of both the mass and angular momentum of the black
strings.
