  We analyze the vortex solution space of the $(2 +1)$-dimensional nonlinear
Dirac equation for bosons in a honeycomb optical lattice at length scales much
larger than the lattice spacing. Dirac point relativistic covariance combined
with s-wave scattering for bosons leads to a large number of vortex solutions
characterized by different functional forms for the internal spin and overall
phase of the order parameter. We present a detailed derivation of these
solutions which include skyrmions, half-quantum vortices, Mermin-Ho and
Anderson-Toulouse vortices for vortex winding $\ell = 1$. For $\ell \ge 2$ we
obtain topological as well as non-topological solutions defined by the
asymptotic radial dependence. For arbitrary values of $\ell$ the
non-topological solutions are bright ring-vortices which explicitly demonstrate
the confining effects of the Dirac operator. We arrive at solutions through an
asymptotic Bessel series, algebraic closed-forms, and using standard numerical
shooting methods. By including a harmonic potential to simulate a finite trap
we compute the discrete spectra associated with radially quantized modes. We
demonstrate the continuous spectral mapping between the vortex and free
particle limits for all of our solutions.
