Abstract: Using the Kirchhoff analogy, the stability and tifurcation of a thinelastic rod in Kovalevskaya case is discussed inthis paper. The analytical integrals of Kirchhoff equations arederived, and a qualitative analysis is conducted for the special caseof a twistless rod when the principal axes of the cross section iscoincident with the Frenet axes.It is shown that the Greenhillcriterion for the straight rod with a circular cross section under theaction of tension and torsion is also valid for the rod with anoncircular cross section in Kovalevskaya case.