  Photon region (PR) in the strong gravitational field is defined as a compact
region where photons can travel endlessly without going to infinity or
disappearing at the event horizon. In Schwarzschild metric PR degenerates to
the two-dimensional photon sphere $r=3r_g/2$ where closed circular photon
orbits are located. The photon sphere as a three-dimensional hypersurface in
spacetime is umbilic (its second quadratic form is pure trace). In Kerr metric
the equatorial circular orbits have different radii for prograde, $r_p$, and
retrograde, $r_r$, motion (where $r$ is Boyer-Lindquist radial variable), while
for $r_p<r<r_r$ the spherical orbits with constant $r$ exist which are no more
planar, but filling some spheres. These spheres, however, do not correspond to
umbilic hypersurfaces. In more general stationary axisymmetric spacetimes not
allowing for complete integration of geodesic equations, the numerical
integration show the existence of PR as well, but the underlying geometric
structure was not fully identified so far. Here we suggest geometric
description of PR in generic stationary axisymmetric spacetimes, showing that
PR can be foliated by {\em partially umbilic hypersurfaces}, such that the
umbilic condition holds for classes of orbits defined by the foliation
parameter. New formalism opens a way of analytic description of PR in
stationary axisymmetric spacetimes with non-separable geodesic equations.
