We apply the perturbative technique which was previously developed in cosmology to now describe the gravitational field of a galaxy with arbitrary rotation curve. The rotation curve fixes the background metric. We use a Schwarzschild background as starting point, however with the correct physical coordinates. Then in the perturbation theory we find two classes of metric perturbations, (i) stationary perturbations in a 3-dimensional vicinity of the central black hole and (ii) non-stationary

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... rbations in an infinitely thin disk. Since the metric perturbations directly give the matter density the theory explains the structure of disk galaxies with a bulge. As the inverse of the standard procedure the theory predicts the matter distribution for a given rotation curve. For small distances the rotation curve is Keplerian, so that the usual Schwarzschild metric and the classical Solar System tests follow.