Lopsidedness is common in spiral galaxies. Often, there is no obvious external cause, such as an interaction with a nearby galaxy, for such features. Alternatively, the lopsidedness may have an internal cause, such as a dynamical instability. In order to explore this idea, we have developed a computer code that searches for self-consistent perturbations in razor-thin disc galaxies and performed a thorough mode-analysis of a suite of dynamical models for disc galaxies embedded in an inert dark-matter halo with varying amounts of rotation and radial anisotropy. Models with two equal-mass counter-rotating discs and fully rotating models both show growing lopsided modes. For the counter-rotating models, this is the well-known counter-rotating instability, becoming weaker as the net rotation increases. The m=1 mode of the maximally rotating models, on the other hand, becomes stronger with increasing net rotation. This rotating m=1 mode is reminiscent of the eccentricity instability in near-Keplerian discs. To unravel the physical origin of these two different m=1 instabilities, we studied the individual stellar orbits in the perturbed potential and found that the presence of the perturbation gives rise to a very rich orbital behaviour. In the linear regime, both instabilities are supported by aligned loop orbits. In the non-linear regime, other orbit families exist that can help support the modes. In terms of density waves, the counter-rotating m=1 mode is due to a purely growing Jeans-type instability. The rotating m=1 mode, on the other hand, grows as a result of the swing amplifier working inside the resonance cavity that extends from the disc center out to the radius where non-rotating waves are stabilized by the models outwardly rising Q-profile.
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