  The HERON project is aimed at studying halos and low surface brightness
details near galaxies. In this second HERON paper we consider in detail deep
imaging (down to surface brightness of ~28 mag/arcsec$^2$ in the r band) for 35
galaxies, viewed edge-on. We confirm a range of low surface brightness features
previously described in the literature but also report new ones. We classify
the observed outer shapes of the galaxies into three main types (and their
prototypes): disc/diamond-like (NGC891), oval (NGC4302), and boxy (NGC3628). We
show that the shape of the outer disc in galaxies does not often follow the
general 3D model of an exponential disc: 17 galaxies in our sample exhibit oval
or even boxy isophotes at the periphery. Also, we show that the less flattened
the outer disc is, the more oval or boxy its structure. Many galaxies in our
sample have an asymmetric outer structure. We propose that the observed
diversity of the galaxy outer shapes is defined by the merger history and its
intensity: if no recent multiple minor or single major merging took place, the
outer shape is diamond-like or discy. On the contrary, interacting galaxies
show oval outer shapes, whereas recent merging appears to transform the outer
shape to boxy.
