Cohomogeneity-one-string integrability of spacetimes

We present a framework for reducing all possible cohomogeneity-one strings, i.e., strings with geometrical symmetry, in a given spacetime to mechanical systems, and for analyzing integrability of the systems. As applications, it is clarified whether the systems of cohomogeneity-one strings in $AdS_5$, $AdS_5 \times S^5$, and $AdS_5 \times T^{p,q}$ are integrable or not. This method may reveal a different type of hidden symmetry of spacetimes.