Symmetry properties and explicit solutions of the generalized Weierstrass system

The method of symmetry reduction is systematically applied to derive several classes of invariant solutions for the generalized Weierstrass system inducing constant mean curvature surfaces and to the associated two-dimensional nonlinear sigma model. A classification of subgroups with generic orbits of codimension one of the Lie point symmetry group for these systems provides a tool for introducing symmetry variables and reduces the initial systems to different nonequivalent systems of ordinary differential equations. We perform a singularity analysis for them in order to establish whether these ordinary differential equations have the Painleve property. These ordinary differential equations can then be transformed to standard forms and next solved in terms of elementary and Jacobi elliptic functions. This results in a large number of new solutions and in some cases new interesting constant mean curvature surfaces are found. Furthermore, this symmetry analysis is extended to include conditional symmetries by subjecting the original system to certain differential constraints. In this case, several new types of nonsplitting algebraic, trigonometric, and hyperbolic multisoliton solutions have been obtained in explicit form. Some physical interpretation of these results in the areas of fluid membranes, string theory, two-dimensionl gravity, and cosmology are given. (C) 2001 American Institute of Physics.