Equigeodesic vectors for homogeneous Riemannian submersions

We study $\pi$-equigeodesic vectors associated with homogeneous fibrations, namely vectors that are geodesic with respect to every homogeneous metric making the projection a Riemannian submersion. We obtain an algebraic criterion characterizing such vectors and apply it to classical flag manifolds and Ledger-Obata spaces. As a framework for this study, given Lie groups $K\subseteq H\subseteq G$ with $H$ and $K$ closed in $G$, and a fixed $G$-invariant metric $g_b$ on $G/H$, we describe the family of $G$-invariant metrics $g$ on $G/K$ for which the natural projection $\pi:(G/K,g)\to(G/H,g_b)$ is a Riemannian submersion. We also give a criterion for the fibers of $\pi$ to be totally geodesic.