Abstract <p>We address the problem of trajectory planning for underactuated Euler-Lagrange systems with a single passive degree of freedom. A widely adopted strategy for this class of systems is the method of virtual holonomic constraints (VHCs). This approach involves imposing a geometric relationship among the generalized coordinates—a virtual holonomic constraint—that must be satisfied along the desired trajectory. By enforcing this constraint, the trajectory planning problem is reduced to finding a particular solution of an auxiliary second-order differential equation, referred to as the reduced dynamics. Most existing work considers regular VHCs, where the reduced dynamics are free of singularities. In contrast, we demonstrate that for certain feasible trajectories, the reduced dynamics necessarily involve singular points—regardless of the choice of generalized coordinates or the parameterization of the VHC. To address this issue, we propose a trajectory planning framework which utilizes the presence of singularities in the reduced dynamics. Our contribution is exemplified through the Pendubot system, in which we reveal a new class of feasible trajectories previously unexplored due to their singular structure. Furthermore, we address the problem of trajectory stabilization for this class and present simulation results illustrating the performance of the proposed closed-loop control scheme.</p> Graphic abstract <p>Periodic motion of the Pendubot, with the second link oscillating near the horizontal position. This behavior corresponds to the brown-highlighted closed trajectory on the phase portrait, which passes through the singular point of node type.</p>