Axially symmetric, stationary gravitational field equations and pseudospherical surfaces

For axially symmetric, stationary gravitational field equations, a new Lax pair of the inverse scattering method is presented from a geometrical point of view. The metric coefficient e sup(2γ)(= -g sub(rho rho)g sub(tt)sup(-1)) is taken as the basic field variable, which satisfies an equation describing pseudospherical surfaces, i.e., surfaces of constant negative Gaussian curvature. The equations for other metric coefficients are also discussed. (author)