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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.
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