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- How can I create a metric describing the space outside a large disk, like an elliptical galaxy?

How can I create a metric describing the space outside a large disk, like an elliptical galaxy? In cylindrical coordinates, ##\phi## would be the angle restricted the the plane, as ##\rho## would be the radius restricted to the plane. I think that if ##z## is suppressed to create an embedding function with just ##\rho## and ##\phi##, it would look very much like the Schwarzschild case, since it is a circle in the plane. But if I suppressed ##\phi##, I think the embedding function of that plane would have arguments of both ##\rho## and ##z##, and it would look more oblong. So I feel like the metric must have these features, but I'm not sure specifically in what arrangment.