Separation of Variables and Hamiltonian Formulation for the Ernst Equation

It is shown that the vacuum Einstein equations for an arbitrary stationary axisymmetric space-time can be completely separated by reformulating the Ernst equation and its associated linear system in terms of a nonautonomous Schlesinger-type dynamical system. The conformal factor of the metric coincides (up to some explicitly computable factor) with the τ function of the Ernst equation in the presence of finitely many regular singularities. We also present a canonical formulation of these results, which is based on a “two-time” Hamiltonian approach, and which opens new avenues for the quantization of such systems.