We consider the standard conic convex problem involving a single second-order cone constraint. No constraint qualification (e.g. Slater) condition is assumed, so that unpleasant features such as non-attainment of the optimum objective value may happen. We first tackle the associated feasibility problem, which amounts to deciding whether the problem is strictly feasible, weakly feasible, weakly infeasible or strongly infeasible. We show that this can be determined by solving a fixed number of linear equality systems, without requiring any iterative process. We then outline how this procedure can be modified in order to solve the original optimization problem. We also discuss the point of view of algorithmic complexity and possible generalizations of this procedure.