This article is largely concerned with the time-discretization of descriptor-variable systems coupled to with complementarity constraints. They are named descriptor-variable linear complementarity systems (DVLCS). More speci cally passive DVLCS with minimal state space representation are studied. The Euler implicit discretization of DVLCS is analysed: the one-step non-smooth problem (OSNSP), that is a generalized equation, is shown to be well-posed under some conditions. Then the convergence of the discretized solutions is studied. Several examples illustrate the applicability and the limitations of the developments.