Deformable structures containing elements in the form of plane wedge and subjected to a temperature or surface load are investigated. The wedge is composed of two isotropic elastic elements. One forming of it slides without friction along the rigid surface. The other forming is loaded. The research is based on the deducing of algebraic equations, that are set directly at the specific point (wedge top). Specific points have the distinctive feature – restrictions redundancy on the state parameters. They represent a system of linear inhomogeneous equations. The properties of this system are studied. Combinations of geometric and material parameters at which a singular point loses its status are identified. Dependencies between design parameters and loads, that causes the incompatibility of the equations, what leads to a singular stress state near the wedge top are detected. Load limitations, ensuring its coherence with all the restrictions at the singular point, are formulated.