We consider the configuration space of 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 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. Our approach is founded on the relationship between the deformation of spatial curves and integrable systems, offering an intriguing example where an integrable system generates an orbit in the space of real solutions to polynomial equations defined by geometric constraints.

翻译：我们研究了满足特定二次方程组的二维球面上点的构型空间。利用椭圆θ函数，我们构造了该构型空间中的周期轨道，并证明这些轨道满足mKdV方程和sine-Gordon方程的半离散类比。所研究的构型空间对应于被称为万花链的连杆机构状态空间，而构造的轨道描述了万花链的特征运动。我们的方法建立在空间曲线变形与可积系统之间的关系基础上，提供了一个引人入胜的实例：可积系统在由几何约束定义的多项式方程实解空间中生成轨道。