A closed-form solution of equilibrium outcomes is generally unavailable in a general equilibrium model with incomplete markets (GEI model). Competitive equilibrium may not be directly computable from the aggregate excess demand functions because they need not be continuous in GEI economies. This paper provides a new characterization of competitive equilibrium in the GEI model by putting into a new perspective the sudden shrinkage of risk-sharing opportunities at a spot price that makes some of the assets redundant and thus, may cause the discontinuity of demand functions. The new characterization is based on the notions of `pre-GEI equilibrium' and `test equilibrium'. Pre-GEI equilibrium outcomes yield a computational equivalent of competitive equilibrium while test equilibrium outcomes may provide existential information on equilibrium. Competitive equilibrium for the GEI model is generically computable as an outcome of pre-GEI equilibrium without resort to any element of the Grassmann manifold.