For an analytic affine control system and an analytic locally controlled invariant distribution DELTA with singularities, which is already invariant under the vector fields corresponding to the controls (in other words is input-insensitive), the problem of finding a feedback that will render DELTA controlled invariant is studied. This can be done if one can solve a degenerate system of partial differential equations (PDEs). The latter can be achieved in the presence of scaling-like symmetries.