We consider a class of nonlinear systems which are absorbable in a linear differential inclusion (LDI), namely whose trajectories are a subset of an LDI. We show than if such an LDI is stabilizable via a linear compensator then a parametrization of all non-linear stabilizing compensators can be derived. The proposed parametrization is simple and the result is constructive. A meaningful application of the parametrization to the switching compensator problem is proposed.