  We present vortex solutions for the homogeneous two-dimensional Bose-Einstein
condensate featuring dipolar atomic interactions, mapped out as a function of
the dipolar interaction strength (relative to the contact interactions) and
polarization direction. Stable vortex solutions arise in the regimes where the
fully homogeneous system is stable to the phonon or roton instabilities. Close
to these instabilities, the vortex profile differs significantly from that of a
vortex in a nondipolar quantum gas, developing, for example, density ripples
and an anisotropic core. Meanwhile, the vortex itself generates a mesoscopic
dipolar potential which, at distance, scales as 1/r^2 and has an angular
dependence which mimics the microscopic dipolar interaction.
