  Using 3-Sasakian reduction techniques we obtain infinite families of new
3-Sasakian manifolds $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$ and
$\scriptstyle{{\cal M}(p_1,p_2,p_3,p_4)}$ in dimension 11 and 15 respectively.
The metric cone on $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$ is a generalization of
the Kronheimer hyperk\"ahler metric on the regular maximal nilpotent orbit of
$\scriptstyle{{\Got s}{\Got l}(3,\bbc)}$ whereas the cone on
$\scriptstyle{{\cal M}(p_1,p_2,p_3,p_4)}$ generalizes the hyperk\"ahler metric
on the 16-dimensional orbit of $\scriptstyle{{\Got s}{\Got o}(6,\bbc)}$. These
are first examples of 3-Sasakian metrics which are neither homogeneous nor
toric. In addition we consider some further $\scriptstyle{U(1)}$-reductions of
$\scriptstyle{{\cal M}(p_1,p_2,p_3)}$. These yield examples of non-toric
3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit
families $\scriptstyle{{\cal O}(\Theta)}$ of compact self-dual positive scalar
curvature Einstein metrics with orbifold singularities and with only one
Killing vector field.
