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

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces