We consider gravitational lensing by a generic extended mass distribution. We represent the static external gravitational field of the lens as a potential via an infinite set of symmetric trace free (STF) moments. We discuss the possibility of determining the physical characteristics of the lens including its shape, orientation and composition via gravitational lensing. To do that, we consider STF multipole moments for several well-known solids with uniform density. We discuss the caustics formed by the point spread function (PSF) of such lenses, and also the view seen by an imaging telescope placed in the strong interference region of the lens. We show that at each STF order, all the bodies produce similar caustics that are different only by their magnitudes and orientations. Furthermore, there is ambiguity in determining the shape of the lens and its mass distribution if only a limited number of moments are used in the model. This result justifies the development of more comprehensive lens models that contain a greater number of multipole moments. At the same time, inclusion of higher multipole moments leads to somewhat limited improvements as their contributions are suppressed by corresponding powers of the small parameter $(R/b)^ell$, where $R$ characterizes the bodys physical size and $b$ is the impact parameter, resulting in a weaker signature from those multipole moments in the PSF. Thus, in realistic observations there will always be some ambiguity in the optical properties of a generic lens, unless the properties of the lens can be determined independently, as in the case of the solar gravitational lens (SGL). Our results are novel and offer new insight into gravitational lensing by realistic astrophysical systems.
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