  We study the spectrum and stationary states in a ring-shaped lattice
potential in the context of ultracold atoms with attractive interatomic
interactions. We determine analytical solutions in the absence of a lattice by
mapping them to those for repulsive interactions, and then we numerically
follow the transformation of those solutions as the lattice is introduced and
strengthened. Several features emerge that are specific to negative
nonlinearity, that include: Soliton branches detaching to create new ground
states; gaps opening up at the bottom of the primary spectral branch; multiple
splitting and rejoining of some branches. We correlate the spectral features
with the behavior of the density and phase of the corresponding eigenstates,
and track them along branches and as various system parameters change. We find
that the phase is sensitive to how a specific point in the spectrum is
approached, particularly relevant at certain persistent gaps in the spectrum.
The symmetry and stability properties are generally found to be opposite of
that found for repulsive interactions.
