  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.
