Photon trapping in static axially symmetric spacetime

Recently, several new characteristics have been introduced to describe null geodesic structure of strong gravitational fields, such as photon regions, transversely trapping surfaces, and some of their generalizations. They provide an alternative and concise way of describing the lensing and shadow features of compact objects without recurring to the complete integration of geodesic equations. Here we test this construction in the case of the static asymptotically flat Weyl metrics where the geodesic equations are nonseparable and cannot be integrated analytically, while the characteristic surfaces can be described in a closed form. Specifically, we consider the gamma-metrics which attracted attention as an alternative to the standard picture of a black hole. We further develop our formalism by introducing a more detailed specification of transversely trapping surfaces in terms of their principal curvatures. Surprisingly, we find certain structures, such as photon regions, previously known only in stationary spacetimes, in the static case without spherical symmetry too. These photon regions can be viewed as photon spheres, "thickened" by oblateness of the metric.