Existence of non-isotropic conjugate points on rank one normal   homogeneous spaces
  We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is variational complete is a rank one symmetric space. 
J. C. Gonz\'alez-D\'avila and A. M. Naveira
math.DG