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The morphology of most of the galaxies hosting counter-rotating components appear undisturbed with no evidence of recent interaction with small satellites or companions of similar size. Indeed, the environment of counter-rotating galaxies does not appear statistically different from that of normal galaxies, as pointed out by Bettoni et al. (2001). They investigated the number, size, and distribution of the faint satellites (to a limiting magnitude B < 21.5) and bright companions (within a searching radius R < 0.6 Mpc and redshift difference ΔV < 600 km s-1) and the large-scale environment of 49 galaxies with counter-rotation and 43 comparison galaxies without counter-rotation.

These findings set constraints on the origin of counter-rotation, because the formation process is required to not affect the present morphology of the host galaxies and the galaxy density of their surrounding regions. Therefore, retrograde gas accretion has to be a smooth and non-traumatic process, major mergers are expected to occur only early in the life of the host galaxy whereas minor mergers may be more recent. However, the relics of merger events like the collisional debris and tidal tails are generally transient and faint structures. Their surviving ages vary from a few hundred Myr to a few Gyr and they have a surface brightness that is typically 25 B-mag arcsec-2 when young and below 27 B-mag arcsec-2 when getting older. The detection of these fine structures requires deep optical imaging, but the comparison of their morphology and kinematics with the results of numerical simulations promises to constrain the epoch and mechanism of the second event (e.g., Corsini et al. 2002, Duc et al. 2011).

As far as the morphology of the host galaxy concerns, no counter-rotating components are detected in late-type spiral galaxies. Three of the few spirals hosting counter-rotating gaseous and/or stellar disks (NGC 3593, NGC 3626, and NGC 4138) belong to the same morphological type, being very early-type spirals (S0/a-Sa) with smooth arms. Their spiral pattern is either defined entirely or dominated by the dust lanes. Indeed, they appear in same section of the The Carnegie Atlas of Galaxies (Plates 72-76; Sandage & Bedke 1994). The suppression of arms in counter-rotating spirals has been recently recognized in high-resolution N-body simulations of multi-armed spiral features triggered through swing amplification by density inhomogeneities (with the mass and lifetime of the order of a typical giant molecular cloud) orbiting the disk (D'Onghia et al. 2013). A survey of a sample of early S0/a and Sa spirals selected to have the spiral pattern traced by dust lanes unveiled the presence of kinematically decoupled gas components but no new case of counter-rotation (Corsini et al. 2003b).

Previous 2-dimensional N-body simulations of disk galaxies with a significant fraction of counter-rotating stars predicted the formation of a stationary and persisting one-arm leading spiral wave (with respect to the corotating stars) due to the two-stream disk instability (Lovelace et al. 1997, Comins et al. 1997). However, the counter-rotating spirals studied so far have intermediate-to-high inclination which makes difficult to identify the presence of a one-armed spiral pattern, whereas kinematic data for low inclined one-arm systems are missing.

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