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.