  An explicit one-parameter Lie point symmetry of the four-dimensional vacuum
Einstein equations with two commuting hypersurface-orthogonal Killing vector
fields is presented. The parameter takes values over all of the real line and
the action of the group can be effected algebraically on any solution of the
system. This enables one to construct particular one-parameter extended
families of axisymmetric static solutions and cylindrical gravitational wave
solutions from old ones, in a simpler way than most solution-generation
techniques, including the prescription given by Ernst for this system. As
examples, we obtain the families that generalize the Schwarzschild solution and
the $C$-metric. These in effect superpose a Levi-Civita cylindrical solution on
the seeds. Exploiting a correspondence between static solutions of Einstein's
equations and Ricci solitons (self-similar solutions of the Ricci flow), this
also enables us to construct new steady Ricci solitons.
