Generalized normal homogeneous spheres $S^{4n+3}$ with greatest connected motion group $Sp(n+1)\cdot U(1)$

We find all (new) $\delta$-homogeneous invariant Riemannian metrics (including the metrics that are not normal homogeneous) on the spheres of dimensions $4n+3$, $n\ge1$, with the greatest Lie group of isometries equal to $Sp(n+1)\times U(1)$ and all homogeneous (non-simply-connected) lens spaces covered by them. All $\delta$-homogeneous Riemannian spaces considered here have positive sectional curvatures and zero Euler characteristic. The answers are found to some previously posed questions.