A system with ideal non-retaining constraint q 1 ⩾ 0 is studied in the neighbourhood of an equilibrium position in which the reaction of the constraint is non-zero. It is assumed that the equilibrium is stable when q 1 0. A family of periodic motions with impacts on the constraint is shown to exist, the period of whose motions tends to zero along with their amplitude. The orbital stability of the periodic motions is studied to a first approximation. It is shown that certain results of KAM theory can be used for non-linear analysis. In particular, conditions are obtained for the equilibrium position of a multidimensional system to be stable for a majority of initial conditions.