Face-on structure of barlenses and boxy bars: an insight from spectral
  dynamics
  Based on the spectral analysis of individual orbits of stars from different
$N$-body models, we show that the face-on morphology of the so-called `face-on
peanut' bars (boxy bars) and barlenses is supported by different types of
orbits. For `face-on peanut' bars, the so-called boxy orbits come to the fore,
and they are responsible for the unusual morphology of the bar in the central
regions. In the models with compact bulges, the bars show a barlens morphology
in their central parts. We found that the barlens supporting orbits come in two
types, one of which gives a square-like shape and the other have a rosette-like
shape in the frame co-rotating with the bar. Such a shape is typical for orbits
around stable loop orbits in nearly axisymmetric potentials only slightly
distorted by the bar. They were already known from some of the previous studies
but their role in barlens shaping was barely investigated. Although quite
simple, the rosette-like orbits are found to be the main building block of a
barlens in our models. The detailed analysis of all bar orbits also allowed us
to distinguish the x2 orbital family and isolate the structure supported by
orbits trapped around this family. The x2 family is well-known, but,
apparently, for the first time in $N$-body models we have revealed the
structure it supports by means of spectral dynamics and highlight its
contribution to the barlens. We found that the x2 family population increases
with an increase in central matter concentration.
