  The Ryu-Takayanagi conjecture connects the entanglement entropy in the
boundary CFT to the area of open co-dimension two minimal surfaces in the bulk.
Especially in AdS(4), the latter are two-dimensional surfaces, and, thus,
solutions of a Euclidean non-linear sigma model on a symmetric target space
that can be reduced to an integrable system via Pohlmeyer reduction. In this
work, we invert Pohlmeyer reduction to construct static minimal surfaces in
AdS(4) that correspond to elliptic solutions of the reduced system, namely the
cosh-Gordon equation. The constructed minimal surfaces comprise a two-parameter
family of surfaces that include helicoids and catenoids in H(3) as special
limits. Minimal surfaces that correspond to identical boundary conditions are
discovered within the constructed family of surfaces and the relevant geometric
phase transitions are studied.
