Integrability of Invariant Geodesic Flows on n-Symmetric Spaces

In this paper, by modifying the argument shift method,we prove Liouville integrability of geodesic flows of normal metrics (invariant Einstein metrics) on the Ledger-Obata $n$-symmetric spaces $K^n\mathrm{/}\text{diag}(K)$, where $K$ is a semisimple (respectively, simple) compact Lie group.