A class of integrable metrics

In four dimensions, the most general metric admitting two commuting Killing vectors and a rank-two Killing tensor can be parametrized by ten arbitrary functions of a single variable. We show that picking a special vierbein, reducing the system to eight functions, implies the existence of two geodesic and share-free, null congruences, generated by two principal null directions of the Weyl tensor. Thus, if the spacetime is an Einstein manifold, the Goldberg-Sachs theorem implies it is Petrov type D, and by explicit construction, is in the Carter class. Hence, our analysis provides a straightforward connection between the most general integrable structure and the Carter family of spacetimes.