This letter provides an exhaustive analysis of all the singularities and minimal cases in cooperative visual-inertial odometry. Specifically, the case of two agents is analysed. As in the case of a single agent and in the case of other computer vision problems, the key of the analysis is the establishment of an equivalence between the cooperative visual-inertial odometry problem and a Polynomial Equation System (PES). In the case of a single agent, the PES consists of linear equations and a single polynomial of second degree. In the case of two agents, the number of second degree equations becomes three and, also in this case, a complete analytic solution can be obtained. The power of the analytic solution is twofold. From one side, it allows us to determine the state without the need of an initialization. From another side, it provides fundamental insights into all the structural properties of the problem. This letter focuses on this latter issue. Specifically, we obtain all the minimal cases and singularities depending on the number of camera images and the relative trajectory between the agents. The problem, when non singular, can have up to eight distinct solutions. The usefulness of this analysis is illustrated with simulations. In particular, we show quantitatively how the performance of the state estimation worsens near a singularity.