  We study some cases when the sectional curvature remains positive under the
taking of quotients by certain nonfree isometric actions of Lie groups. We
consider the actions of the groups $S^1$ and $S^3$ such that the quotient space
can be endowed with a smooth structure using the fibrations
$S^3/S^1{\simeq}S^2$ and $S^7/S^3\simeq S^4$. We prove that the quotient space
carries a metric of positive sectional curvature, provided that the original
metric has positive sectional curvature on all 2-planes orthogonal to the
orbits of the action.
