  We introduce a family of models for magnetic skyrmions in the plane for which
infinitely many solutions can be given explicitly. The energy defining the
models is bounded below by a linear combination of degree and total vortex
strength, and the configurations attaining the bound satisfy a first order
Bogomol'nyi equation. We give explicit solutions which depend on an arbitrary
holomorphic function. The simplest solutions are the basic Bloch and N\'eel
skyrmions, but we also exhibit distorted and rotated single skyrmions as well
as line defects, and configurations consisting of skyrmions and anti-skyrmions.
