we refer to Sect. \[cmsec\] and the appendix below. Also, this condition is satisfied in numerous concrete situations, for example when the cohomogeneity of the action is zero or one, as it was pointed out in [@Go-Ma].

Cohomogeneity-one actions are special cases of hyperpolar actions. Recall that an isometric action of a compact connected Lie group on a Riemannian manifold is [*polar*]{} if there exists a submanifold which is intersected by each orbit of the action, the orbit being orthogonal to the submanifold at each intersection point. If the submanifold can be chosen to be flat relative to the induced Riemannian metric, we say that the action is [*hyperpolar*]{}. In this paper we will be interested in the case when the manifold on which the group acts is a Riemannian symmetric space of compact type, that is, a quotient $G/H$, where $G$ is a compact connected semisimple Lie group and $H$ a closed subgroup such that $G^\sigma_0 \subset H \subset G^\sigma$, where $\sigma$ is an involutive automorphism of $G$, $G^\sigma$ its fixed point set and $G^\sigma_0$ the identity component of the latter group. Polar actions on compact symmetric spaces have been extensively investigated by many authors, see for instance [@HPTT1], 