  We study the landscape of solutions of the coherent quantum states in a ring
shaped lattice potential in the context of ultracold atoms with an effective
positive nonlinearity induced by interatomic interactions. The exact analytical
solutions in the absence of lattice are used as a starting point and the
transformation of those solutions is mapped as the lattice is introduced and
strengthened. This approach allows a simple classification of all the solutions
into states with periods commensurate/incommensruate with the lattice period
and those with/without nodes. Their origins are traced to the primary
dispersion curve and the swallowtail branches of the lattice-free spectrum. The
commensurate states tend to remain delocalized with increasing lattice depth,
whereas the incommensurate ones may be localized. The symmetry and stability
properties of the solutions are examined and correlated with branch energies.
The crucial importance of rotation is highlighted by its utility in
continuously transforming solutions and accessing in a finite ring with a few
sites the full spectrum of nonlinear Bloch waves on an infinite lattice.
