The problem of lateral stability of faired underwater towing cables was the physical problem motivating the author to study mathematically the lateral buckling of nonlinearly elastic beams. The equations are derived by means of a special Cosserat theory. The beams together with their loads have a plane of symmetry. As first step the unbuckled, basic (”trivial”) solutions which are planar solutions are studied which, however, by no means are trivial at all. About these basic solutions the rod equations are linearized and the corresponding eigenvalue problem is solved. Only divergence bifurcations where buckled configurations exist are treated, whereas the possibility of Hopf bifurcations leading to a flutter instability are only indicated. Global results are given to describe all bifurcated solutions.