The criterion for the creation of a flux phase state on a ring is studied by the Bethe-ansatz method. First we calculate exactly the spectrum of fermions located on a ring in a transverse magnetic field in the framework of models associated with the model of Heisenberg-Ising spin chains, i.e. we take into account only the next neighbor interaction. We find universality in the behavior of interacting fermions on the ring in the free fermion case in this model and prove the existence of the flux phase state on a ring. The physical reason for this effect is the appearance of a statistical flux at an even number of fermions on a ring. We show that a flux phase state is also the result of the magnetic Jahn-Teller effect. The conclusion does not depend on the interaction between the fermions. A relation between different oscillation phenomena on the ring and a hypothetic polymer magnet is proposed.