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

Static Axisymmetric Einstein Equations in Vacuum: Symmetry, New Solutions and Ricci Solitons