  We study an axisymmetric metric satisfying the Petrov type D property with
some additional ansatze, but without assuming the vacuum condition. We find
that our metric in turn becomes conformal to the Kerr metric deformed by one
function of the radial coordinate. We then study the gravitational-wave
equations on this background metric in the case that the conformal factor is
unity. We find that under an appropriate gauge condition, the homogeneous wave
equations admit the separation of the variables, which is also helpful for
solving the nonhomogeneous equations. The resultant ordinary differential
equation for the radial coordinate gives a natural extension of the Teukolsky
equation.
