Integrability of strings with a symmetry in the minkowski spacetime

We investigate the integrability of Nambu-Goto strings with a cohomogeneity-one symmetry in the Minkowski spacetime. By virtue of the symmetry, the equations of motion reduce to the geodesic equations in a three dimensional orbit space. The orbit space inherits the Killing vectors from the Minkowski spacetime, which help us to integrate the geodesic equations. The cohomogeneity-one strings are fallen into seven families (Type I ∼ VII) in the Minkowski spacetime. We find that the strings of Type I ∼ VI are integrable due to the existence of two or more inherited Killing vectors. We also find that the strings of Type VII are integrable due to the existence of an inherited Killing vector and a Killing tensor.