Geometrical aspects of the complex eikonal equations associated to the generalized Weierstrass representation

Certain geometrical aspects of a Laplace-Beltrami equation subjected to a c omplex eikonal equation are studied. It is. shown that all principal curvat ures of a solution surface are constant for this system. To illustrate the theoretical considerations, we look for a class of solutions of an overdete rmined system composed of the sigma model and a system of complex eikonal e quations. The connection between the sigma model and the generalized Weiers trass system for inducing constant mean curvature surfaces allows us to con struct special classes of solutions for classical configurations of strings .