Compressible binormal equation for membranes

Determine an interesting, possibly integrable in lower dimensions, analog of the binormal equation for membranes in the compressible setting of augmented triples on a Riemannian manifold.

Background

The paper develops a generalized Marsden–Weinstein symplectic structure on singular 1-form densities, represented by augmented triples $(\Gamma,\rho,\alpha)$, as a compressible-fluid counterpart of the classical Marsden–Weinstein structure for incompressible vortex membranes. In the incompressible setting, the length or volume Hamiltonian generates the binormal or skew-mean-curvature evolution.

For augmented triples, the authors explain that a seemingly natural volume Hamiltonian is insufficient because the generalized Hamiltonian structure depends on first jets of the Hamiltonian, equivalently second jets of the associated vector field near $\Gamma$. They therefore leave unresolved the construction of a suitable compressible analog of binormal dynamics, including the possibility of integrable cases in low dimensions.