A characterization of carter's separable space-times

In a space-time admitting a two-parameter Abelian isometry group, and a quadratic Killing tensor with the eigenvalues (, , , ) and vanishing Lie derivatives with respect to the Killing vectors, we construct a canonical coordinate system. The isometry group acts orthogonally transitively. The Hamilton-Jacobi equation is separable. We give a necessary and sufficient condition for the separability of the Klein-Gordon equation. We obtain Carter's space-times with completely separable Klein-Gordon equation.