  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.
