  We study static cylindrically symmetric vacuum solutions in Weyl coordinates
in the context of the metric f(R) theories of gravity. The set of the modified
Einstein equations is reduced to a single equation and it is shown how one can
construct exact solutions corresponding to different $f(R)$ models. In
particular the family of solutions with constant Ricci scalar ($R=R_{0}$) is
found explicitly which, as a special case (R=0), includes the exterior
spacetime of a cosmic string. Another new solution with constant, non-zero
Ricci scalar is obtained and its possible relation to the Linet-Tian solution
in general relativity is discussed.
