The Ernst equation of General Relativity for stationary axially symmetric fields can be written as $$(E+\bar{E})\Delta E=2\nabla E\cdot\nabla E$$where $E=F+iG$ and $\bar{E}=F-iG$.
In cylindrical coordinates $(r,\phi,z)$, this equation can be written as$$F(F_{rr}+F_{zz}+F_{r}/r)=F_{r}^2+F_{z}^2-G_{r}^2-G_{z}^2\\F(G_{rr}+G_{zz}+G_{r}/r)=2(F_{r}G_{r}+F_{z}G_{z})$$
The Lie point symmetries of these PDEs are for the independent variables$$\frac{\partial}{\partial z}$$ $$r\frac{\partial}{\partial r}+z\frac{\partial}{\partial z}$$ and for the dependent variables$$\frac{\partial}{\partial G}$$ $$F\frac{\partial}{\partial F}+G\frac{\partial}{\partial G}$$ $$FG\frac{\partial}{\partial F}+1/2(G^2-F^2)\frac{\partial}{\partial G}$$ All of the solutions that I can find come from the Ernst equation with the dependent variables F and G transformed by the Ehlers transformation with the following form given by B. K. Harrison in his 2004 article "Applications of Symmetry to General Relativity" found as a pdf on the web:$$E'=\frac{(aE+ib)}{(icE+d)}$$where a,b,c,d are real constants and $E=F+iG$. He states that this transformation is obtained when the above 3 symmetries of the dependent variables are combined and exponentiated.
So here is where I need help because the text books that I have are not much help. Can someone show me how to do this, i.e. combine and exponentiate those symmetries to get that transformation? Or alternatively, is there some text or article that derives this, and is accessible to someone with an undergraduate level of expertise (in physics)?
The problem is answered (without the use of Lie symmetries) in the free pdf article by Yishi Duan on page 9:
http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-3265.pdf