  To generate the standard microlensing light curve one assumes that the
relative motion of the source, the lens, and the observer is linear. In
reality, the relative motion is likely to be more complicated due to
accelerations of the observer, the lens and the source. The simplest
approximation beyond the linear-motion assumption is to add a constant
acceleration. Microlensing light curves due to accelerations can be symmetric
or asymmetric depending on the angle between the acceleration and the velocity.
We show that it is possible that some of the previously reported shorter
marginal parallax events can be reproduced with constant-acceleration models,
while the longer, multi-year parallax events are ill-fitted by such models. We
find that there is a generic degeneracy inherent in constant-acceleration
microlensing models. We also find that there is an equivalent degeneracy in
parallax models, which manifests itself in short-duration events. The
importance of this new parallax degeneracy is illustrated with an example,
using one of these marginal parallax events. Our new analysis suggests that
another of these previously suspected parallax candidate events may be
exhibiting some weak binary-source signatures. If this turns out to be true,
spectroscopic observations of the source could determine some parameters in the
model and may also constrain or even determine the lens mass. We also point out
that symmetric light curves with constant accelerations can mimic blended light
curves, producing misleading Einstein-radius crossing time-scales when fitted
by the standard `blended' microlensing model; this may have some effect on the
estimation of optical depth.
