A Class of Integrable Metrics II

Starting with a subclass of the four-dimensional spaces possessing two commuting Killing vectors and a non-trivial Killing tensor, we fully integrate Einstein's vacuum equation with a cosmological constant. Although most of the solutions happen to be already known, we have found a solution that, as far as we could search for, has not been attained before. We also characterize the geometric properties of this new solution, it is a Kundt spacetime of Petrov type II possessing a null Killing vector field and an isometry algebra that is three-dimensional and abelian. In particular, such solution becomes a pp-wave spacetime when the cosmological constant is set to zero.