The complex eikonal equation in the three space dimensions is considered. We
show that apart from the recently found torus knots this equation can also
generate other topological configurations with a non-trivial value of the
$\pi_2(S^2)$ index: braided open strings as well as hedgehogs. In particular,
cylindric strings i.e. string solutions located on a cylinder with a constant
radius are found. Moreover, solutions describing strings lying on an arbitrary
surface topologically equivalent to cylinder are presented. We discus them in
the context of the eikonal knots. The physical importance of the results
originates in the fact that the eikonal knots have been recently used to
approximate the Faddeev-Niemi hopfions.