  We study the integral transform over a general family of broken rays in
$\mathbb{R}^2$. It is natural for broken rays to have conjugate points, for
example, when they are reflected from a curved boundary. If there are conjugate
points, we show that the singularities cannot be recovered from local data and
therefore artifacts arise in the reconstruction. We apply these conclusions to
two examples, the V-line Radon transform and the parallel ray transform. In
each example, a detailed discussion of the local and global recovery of
singularities is given and we perform numerical experiments to illustrate the
results.
