I've been checking the pages related to "Kerr metric" and "Frame dragging". I suppose these from wiki confirm that rotation affects the distortion.
"A stationary field is one that is in a steady state, but the masses causing that field may be non-static, rotating for instance."
"They predicted that the rotation of a massive object would distort the spacetime metric, making the orbit of a nearby test particle precess. This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation."