The problem of plane deformation of a thin elastic inextensible rod of length L supported by a smooth inclined wall and the stability of a compressed rod under restrictions on the displacement were discussed. The problem is distinguished by the presence of a wall preventing the rod from deflection in one of the transverse directions. The deflection in the linear approximation is assumed to be small and the compression force is considered equal to its value in the unstrained state. The linear approximation allows finding only unstable equilibria and it was also demonstrated that the solutions constructed in the linear approximation are unstable as well. It was found that the calculation of the value of A reduced to the determination of the minimum eigenvalue of the linear boundary value problem. It was shown by calculations that either the stability conditions were violated or the condition at the reaction is positive.