Photon and transversely trapping surfaces in stationary spacetime

The photon surface (PS) is defined as closed timelike hypersurface in spacetime such that any geodesic initially tangent to its spatial section remains in it forever. The transversely trapping surface (TTS) is such that geodesics can leave this spatial section, but only in one direction - inside. Both surfaces are important characteristics of a strong gravitational field of black holes, wormholes and naked singularities associated with their imaging properties. We analyze the PS and the TTS using a number of examples with an emphasis on their existence in static spaces that do not possess spherical symmetry, as well as in stationary spaces. Solutions are investigated in which classical photon surfaces disappear, but there are TTS and their generalizations. Axially symmetric metrics with rotation and with naked singularities, such as the Zipoy-Vourhis solution, are considered.