We prove that a compact Riemannian 3-symmetric space is globally symmetric if every Jacobi field along a geodesic vanishing at two points is the restriction to that geodesic of a Killing field induced by the isotropy action or, in particular, if the isotropy action is variationally complete.
"Isotropic Jacobi fields on compact 3-symmetric spaces." J. Differential Geom. 83 (2) 273 - 288, October 2009. https://doi.org/10.4310/jdg/1261495332