  Non-linear sigma models defined on symmetric target spaces have a wide set of
applications in modern physics, including the description of string propagation
in symmetric spaces, such as AdS or dS, or minimal surfaces in hyperbolic
spaces. Although it is difficult to acquire solutions of these models, due to
their non-linear nature, it is well known that they are reducible to integrable
systems of the family of the sine- or sinh-Gordon equation. In this study, we
develop a method to invert Pohlmeyer reduction for elliptic solutions of the
reduced system, implementing a relation between NLSM solutions and the
eigenstates of the n = 1 Lame problem. This method is applied to produce a
family of classical string solutions in AdS(3), which includes the spiky
strings, as well as hoop string solutions with singular evolution of their
angular velocity and radius, which are interesting in the framework of
holographic dualities. Furthermore, application of this method produces a wide
family of static minimal surfaces in AdS(4), which includes helicoids and
catenoids, and which are interesting in the framework of the Ryu-Takayanagi
conjecture and the understanding of the emergence of gravity as an entropic
force related to quantum entanglement statistics. The developed formalism
allows the study of the area of the minimal surfaces and geometric phase
transitions between them, which are relevant to confinement-deconfinement phase
transitions.
