  A Riemannian manifold $(M,\rho)$ is called Einstein if the metric $\rho$
satisfies the condition $\Ric (\rho)=c\cdot \rho$ for some constant $c$. This
paper is devoted to the investigation of $G$-invariant Einstein metrics with
additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As
a consequence, we obtain new invariant Einstein metrics on some Stiefel
manifolds $SO(n)/SO(l)$, and on the symplectic analogues $Sp(n)/Sp(l)$.
Furthermore, we show that for any positive integer $p$ there exists a Stiefel
manifold $SO(n)/SO(l)$ and a homogenous space $Sp(n)/Sp(l)$ which admit at
least $p$ SO(n) (resp. $Sp(n)$)-invariant Einstein metrics.
