Einstein's vacuum spaces with an axial symmetry

Four classes of solutions have been obtained for equations Rsub(αβ)=-2 sigma;sub(α)sup(sigma);β,gsup(αβ)sigma;sub(α)β=0 in a three-dimensional space with the metric of gsub(α)βdxαdxsup(β) of the signature (+--). The equations are equivalent to the Einstein equations Rsub(ij)=0 in a vacuum for the metric dssup(2)--esup(2 sigma)gsub(αβ)dxsup(α)dxsup(β)-esup(-2 sigma)xdxsup(2sup(3)). Metric ds2 allows the axial symmetry and that relative to the reflection of Xsup(3')=-Xsup(3)