Secondary Bifurcation Near a Double Eigenvalue

General conditions are formulated under which secondary bifurcation is rigorously established for a family of bifurcation problems depending continuously on a real auxiliary parameter. With more specific conditions, it is shown that, although the presence of secondary bifurcation renders the problem a priori degenerate, a full local bifurcation analysis is still possible. The results of this paper demonstrate the prime importance of symmetry (or more generally, invariance) to the mechanism by which secondary bifurcation points are created as the auxiliary parameter is varied.