Integral equation in the theory of phase transitions in the system of magnetic rods

The nonlinear integral equation for the orientational distribution function (ODF), describing anisotropic nematic ordering in a system of magnetic rods, is investigated. The classification of bifurcation points is presented and their asymptotics are found for highly elongated rods with the small and large magnetic moments. An algorithm for finding the ODF, near to isotropic, is developed in the neighborhood of bifurcation points. In the limiting cases of small and large magnetic moments of rods the found solutions (ODF) have the left direction of bifurcation. However, in the intermediate region of values of the magnetic moments the solutions having the right direction of bifurcation exist along with solutions having the left direction of bifurcation.