One of the drawbacks of the eXtended Finite Element Method and similar approaches like the Generalized Finite Element Method is the problem of illconditioning of the related systems of equations in the solution stage. This occurs for example in Heaviside function enrichments when the discontinuity is close to discretisation nodes but also for non-linear enrichment functions used in conjunction to geometric enrichment domains. In the present work the motivation of illconditioning is analyzed to derive a novel methodology for stabilization, based on setting proper constraints for the variables. This methodology does not impact on the initial formulation nor in the element stiffness computation, so that it is very effective. Results are analyzed in 1D and 3D to show its performance and properties