This article is based on earlier author's works on the behavior of mechanical systems with a singular configuration space. We obtained exact analytical expressions for the constraints providing multi-mode oscillations of a double mathematical pendulum. It was found that there are infinitely many modes of oscillations determined by the degree of degeneracy at a singular point in the configuration space of the pendulum. The problem of a double pendulum motion with an elliptical constraint, posed by the author earlier, received a complete solution, both theoretical and experimental. For such a constraint, there are exactly two different oscillation modes: Symmetric and asymmetric. Each mode can be realized by selecting the system parameters and is completely determined by the smoothness of the trajectory in the configuration space with a singularity. In the study we used only standard tools of classical analytical mechanics.