  For an orientable surface of finite type equipped with a flat metric with
holonomy of finite order q, the set of maximal embedded cylinders can be empty,
non-empty, finite, or infinite. The case when q < 3 is well-studied as such
surfaces are (semi-)translation surfaces. Not only is the set always infinite,
the core curves form an infinite diameter subset of the curve complex. In this
paper we focus on the case q > 2 and construct examples illustrating a range of
behaviors for the embedded cylinder curves. We prove that if q > 2 and the
surface is fully punctured, then the embedded cylinder curves form a finite
diameter subset of the curve complex. The same analysis shows that the embedded
cylinder curves can only have infinite diameter when the metric has a very
specific form. Using this we characterize precisely when the embedded cylinder
curves accumulate on a point in the Gromov boundary.
