  A connection between the dynamics of a sine-Gordon chain and a certain static
membrane folding problem was recently found. The one-dimensional membrane
profile is a cross-section of the position-time sine-Gordon amplitude profile.
Here we show that when one system is embedded in a higher-dimensional system in
this way, obvious symmetries in the larger system can lead to nontrivial
symmetries in the embedded system. In particular, a thin buckled membrane on a
fluid substrate has a continuous degeneracy that interpolates between a
symmetric and an antisymmetric fold. We find the Hamiltonian generator of this
symmetry and the corresponding conserved momentum by interpreting the simple
translational symmetries of the sine-Gordon chain in terms of the embedded
coordinates. We discuss possible extensions to other embedded dynamical
systems.
