We consider a Bose-Einstein condensate with repulsive interactions confined in a one-dimensional ring where a Dirac δ is rotating at constant speed. The spectrum of stationary solutions in the δ comoving frame is analyzed in terms of the nonlinear coupling, δ velocity, and δ strength, which may take positive and negative values. It is organized into a set of energy levels conforming a multiple swallowtail structure in parameter space, consisting of bright solitons, gray and dark solitonic trains, and vortex states. Analytical expressions in terms of Jacobi elliptic functions are provided for the wave functions and chemical potentials. We compute the critical velocities and perform a Bogoliubov analysis for the ground state and first few excited levels, establishing possible adiabatic transitions between the stationary and stable solutions. A set of adiabatic cycles is proposed in which gray and dark solitons, and vortex states of arbitrary quantized angular momenta, are obtained from the ground state by setting and unsetting a rotating δ. These cycles are reproduced by simulations of the time-dependent Gross-Pitaevskii equation with a rotating Gaussian link.