We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of M\"uller's conjecture \cite{Muller} concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of the primary branch connections that occur in this problem. We give a reformulation of M\"uller's conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a ``loop'' structure that characterizes $(k,3k)$ bifurcations.
