also investigated in the literatures[@VLS; @FSZH; @Anderson], it is worth clarifying physical properties of stationary rotating strings completely because the stationary rotating strings have a rich variety of configuration even in Minkowski spacetime.

A stationary string is an example of cohomogeneity-one string. The world surface of cohomogeneity-one string is tangent to a Killing vector field of a target space, equivalently, the world surface is foliated by the orbits of one-parameter group of isometry. All possible cohomogeneity-one strings in Minkowski spacetime are classified into seven families[@IshiharaKozaki]. Stationary rotating strings in Minkowski spacetime belong to one family of them. Solving equations of motion for cohomogeneity-one string is reduced to finding geodesic curves in a three-dimensional space with the metric which is determined by the Killing vector[@FSZH; @IshiharaKozaki].

We would find an analogy between the system of cohomogeneity-one string and the system of Bianchi cosmologies. In the Bianchi cosmologies, universe is foliated by homogeneous time slices, i.e., spacetime is cohomogeneity-one. The dynamics of Bianchi cosmologies is regarded as the one of a relativistic particle. Similarly, the dynamics of each family of cohomogeneity-one string can be identified by the one of a particle moving in a curved space specified by the geometrical symmetry of 