  In analogy with the standard derivation of the Schwarzschild solution, we
find all static, cylindrically symmetric solutions of the Einstein field
equations for vacuum. These include not only the well known cone solution,
which is locally flat, but others in which the metric coefficients are powers
of the radial coordinate and the space-time is curved. These solutions appear
in the literature, but in different forms, corresponding to different
definitions of the radial coordinate. Because all the vacuum solutions are
singular on the axis, we attempt to match them to "interior" solutions with
nonvanishing energy density and pressure. In addition to the well known "cosmic
string" solution joining on to the cone, we find some numerical solutions that
join on to the other exterior solutions.
