On solutions of the shape equation for membranes and strings

Classical configurations of strings and shapes of membranes in equilibrium are defined by a nonlinear equation. It is shown that this equation has a simple form in terms of the inverse mean curvature and density of squared mean curvature. Broad variety of its solutions (in particular, kink, vortex and solitons) and corresponding possible shapes are given. A new type of degeneracy of membrane shapes and string configurations via the integrable Veselov-Novikov equation is discussed.