  In this paper we develop new methods of study of generalized normal
homogeneous Riemannian manifolds. In particular, we obtain a complete
classification of generalized normal homogeneous Riemannian metrics on spheres.
We prove that for any connected (almost effective) transitive on $S^n$ compact
Lie group $G$, the family of $G$-invariant Riemannian metrics on $S^n$ contains
generalized normal homogeneous but not normal homogeneous metrics if and only
if this family depends on more than one parameters. Any such family (that
exists only for $n=2k+1$) contains a metric $g_{\can}$ of constant sectional
curvature 1 on $S^n$. We also prove that $(S^{2k+1}, g_{\can})$ is
Clifford-Wolf homogeneous, and therefore generalized normal homogeneous, with
respect to $G$ (excepting the groups $G=SU(k+1)$ with odd $k+1$). The space of
unit Killing vector fields on $(S^{2k+1}, g_{\can})$ from Lie algebra
$\mathfrak{g}$ of Lie group $G$ is described as some symmetric space (excepting
the case $G=U(k+1)$ when one obtains the union of all complex Grassmannians in
$\mathbb{C}^{k+1}$).
