  We study the integrability of the equations of motion for the Nambu-Goto
strings with a cohomogeneity-one symmetry in Minkowski spacetime. A
cohomogeneity-one string has a world surface which is tangent to a Killing
vector field. By virtue of the Killing vector, the equations of motion can be
reduced to the geodesic equation in the orbit space. Cohomogeneity-one strings
are classified into seven classes (Types I to VII). We investigate the
integrability of the geodesic equations for all the classes and find that the
geodesic equations are integrable. For Types I to VI, the integrability comes
from the existence of Killing vectors on the orbit space which are the
projections of Killing vectors on Minkowski spacetime. For Type VII, the
integrability is related to a projected Killing vector and a nontrivial Killing
tensor on the orbit space. We also find that the geodesic equations of all
types are exactly solvable, and show the solutions.
