Conformally flat sources for the Linet-Tian spacetime

We investigate the matching, across cylindrical surfaces, of static cylindrically symmetric conformally flat spacetimes with a cosmological constant $\mathrm{\Lambda }$, satisfying regularity conditions at the axis, to an exterior Linet-Tian spacetime. We prove that for $\mathrm{\Lambda }\le 0$ such matching is impossible. On the other hand, we show through simple examples that the matching is possible for $\mathrm{\Lambda }>0$. We suggest a physical argument that might explain these results.