The dynamics of a pair of identical Josephson junctions coupled through a shared purely capacitive load are governed by a two-parameter system of two second-order nonlinear ordinary differential equations. Numerical simulations have shown that this system possesses many different running and periodic solutions. Continuation studies using AUTO indicate that many of these solution branches are generated by a codimension-2 connection which occurs at a particular parameter point. In this paper, we first describe these calculations in detail. We then study a general two-parameter system whose properties reflect some of those found in our numerical studies of the Josephson junction system. In particular, our model system is assumed to possess an appropriate codimension-2 connection, and we prove that its unfolding generates a large variety of codimension-1 connection curves. These results, combined with the particular symmetry and periodicity properties of the junction equations, account for all of the numerically observed solution branches. Indeed, the theoretical analysis predicted the existence of branches which were not initially observed, but which were subsequently found.