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.
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