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.