Axially symmetric gravitational fields. III. New methods of generating stationary vacuum Einstein fields

The Einstein equation for stationary axially-symmetric vacua reduces to a system of nonlinear partial differential equations of second order for two unknowns. The system can be further reduced to the well-known Ernst equation for a single complex function, which has been studied by many authors. The authors of the present work return to the system of two equations and propose two distinct substitutions which make it possible to transform it into a system of linear partial differential equations of first order. The system of equations obtained in this way can be solved and yields some formal solutions which can further be used for generating solutions of the Einstein equation. The authors show that the well-known Kerr solution can also be obtained by one of their generating methods. par {For Part II see [Ts. I. Gutsunaev and N. Sh. Hassan, Gravit. Cosmol. {bf 9} (2003), no.~3, 121--133; [msn] MR2020880 (2004k:83031) [/msn]].}