  We study spacetimes that lead to a separable Klein-Gordon equation in a
general dimension. We introduce an ansatz for the metric in higher dimensions
motivated by analogical work by Carter in four dimensions and find solutions of
the Klein-Gordon equation. For such a metric we solve the Einstein equations
and regain the Kerr-NUT-(A)dS spacetime as one of our results. Other solutions
lead to the Einstein-K\"ahler metric of a Euclidean signature. Next we
investigate a warped geometry of two Klein-Gordon separable spaces with a
properly chosen warped factor. We show that the resulting metric leads also to
a separable Klein--Gordon equation and we find the corresponding solutions.
Finally, we solve the Einstein equations for the warped geometry and obtain new
solutions.
