Explicit determination of certain periodic motions of a generalized two-field gyrostat

The case of motion of a generalized two-field gyrostat found by V.V.Sokolov and A.V.Tsiganov is known as a Liouville integrable Hamiltonian system with three degrees of freedom. We find a set of points at which the momentum map has rank 1. This set consists of special periodic motions which correspond to the singular points of a bifurcation diagram on an iso-energetic surface. For such motions the phase variables can be expressed in terms of algebraic functions of a single auxiliary variable. These algebraic functions satisfy a differential equation integrable in elliptic functions of time. It is shown that the corresponding points in the three-dimensional space of the constants of the integrals belong to the intersection of two sheets of the discriminant surface of the Lax curve.