We consider the configuration space of ordered points on the two-dimensional sphere that satisfy a specific system of quadratic equations. We construct periodic orbits in this configuration space using elliptic theta functions and show that they simultaneously satisfy semi-discrete analogues of mKdV and sine-Gordon equations. The configuration space we investigate corresponds to the state space of a linkage mechanism known as the Kaleidocycle, and the constructed orbits describe the characteristic motion of the Kaleidocycle. A key consequence of our construction is the proof that Kaleidocycles exist for any number of tetrahedra greater than five. Our approach is founded on the relationship between the deformation of spatial curves and integrable systems, offering an intriguing example where an integrable system is explicitly solved to generate an orbit in the space of real solutions to polynomial equations defined by geometric constraints.

翻译：我们考虑了二维球面上满足特定二次方程组的排序点构成的位形空间。利用椭圆theta函数在该位形空间中构造了周期轨道，并证明了这些轨道同时满足mKdV方程和正弦-戈登方程的半离散类比。所研究的位形空间对应于被称为万花环的连杆机构的状态空间，而构造的轨道描述了万花环的特征运动。我们构造的一个关键结论是证明了当四面体数量大于五时，万花环必定存在。该方法基于空间曲线形变与可积系统之间的关系，提供了一个引人入胜的实例，即可积系统被显式求解，从而在由几何约束定义的多项式方程实解空间中生成轨道。