  We find the angular Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states (or the
twisted kink crystals) in which a phase and an amplitude of a pair potential
modulate simultaneously in a quasi-onedimensional superconducting ring with a
static Zeeman magnetic field applied on the ring and static Aharonov-Bohm
magnetic flux penetrating the ring. The superconducting ring with magnetic flux
produces a persistent current, whereas the Zeeman split of Fermi energy results
in the spatial modulation of the pair potential. We show that these two
magnetic fields stabilize the FFLO phase in a large parameter region of the
magnetic fields. We further draw the phase diagram with the two kinds of
first-order phase transitions; one corresponds to phase slips separating the
Aharonov-Bohm magnetic flux, and the other separates the number of peaks of the
pair amplitude for the Zeeman magnetic field.
