The motion of a material point near a singularity of the type of two tangent surfaces is considered. The surfaces are located symmetrically with respect to a common tangent plane and have a common axis of rotation. First, a model of motion for holonomic mechanics is considered. It is shown that only trajectories in a fixed plane containing the axis of rotation of the surfaces can pass through a singular point. At the point of contact, dynamic uncertainty arises, since the trajectory has several possible branches of motion. To study the motion of a material point near a singularity of the type of double tangent paraboloid, a model of the implementation of holonomic constraints through an elastic potential with a large stiffness parameter, or a ``stiff potential'', is considered. The potential must vanish on the manifold with singularities and must be strictly positive outside it. For a model with a stiff potential, it also turns out that only trajectories in a fixed plane containing the axis of rotation of the paraboloids can pass through a singular point. Numerical modeling of the dynamics was done. It was found that the trajectories of a system with a stiff potential can qualitatively differ from the trajectories of the corresponding holonomic system. A holonomic system instantly passes a geometric singularity, moving with a non-zero velocity. A system with a stiff potential can move in a singular region for a finite time, resulting in rapid changes in the direction of the velocity vector. In real mechanical systems, this type of motion can lead to breakdowns or instability of trajectories.