  We carry out the canonical quantization of the Levi-Civit\`a family of static
and cylindrical solutions. The reduced phase space of this family of metrics is
proved to coincide with that corresponding to the Kasner model, including the
associated symplectic structures, except for that the respective domains of
definition of one of the phase space variables are not identical. Using this
result, we are able to construct a quantum model that describes spacetimes of
both the Levi-Civit\`a and the Kasner type, and in which the three-dimensional
spatial topology is not uniquely fixed. Finally, we quantize to completion the
subfamily of Levi-Civit\`a solutions which represent the exterior gravitational
field of a straight cosmic string. These solutions are conical geometries,ie,
Minkowski spacetime minus a wedge. The quantum theory obtained provides us with
predictions about the angular size of this wedge.
