A B S TRACT. Two exact axially symmetric solutions of the gravitational field equations, which depend on a number of arbitrary real constants, are derived.
A problem which has received much attention in recent years is that of finding solutions of the axially symmetric Einstein equations [1,2]. The efforts are mainly concentrated on finding new solutions of the Ernst equation [3], either directly from the equation or by applying Biicklund transformations to known solutions [2, 4 -6 ] . Another method is based on the fact that, in some cases, the equations, which the self-dual Yang-Mills fields in the R-gauge have to satisfy, become equivalent to the Ernst equation [7]. Therefore, solutions of the Ernst equation can be obtained from certain types of self-dual fields either directly or with the help of the B~cklund transformations of these fields [8]. Sometimes however, as explained below, even though the field are not of the appropriate type, or the Backlund transformations alone do not lead to solutions of the Ernst equation, solutions of this equation are obtained directly if we use Theorem 3 of the present work. Two solutions are found in this way, which depend on a number of arbitrary real constants. The solutions are not asymptotically fiat.
To find these solutions we consider a self-dual SU(2) gauge field b~, p = 1,2 .... 4, a = 1,2, 3 in Euclidean space, which is independent of one of the x u variables, say the variable x4, and we introduce the variables y, j and z by the relations