  We study existence of invariant Einstein metrics on complex Stiefel manifolds
$G/K = \SU(\ell+m+n)/\SU(n) $ and the special unitary groups $G =
\SU(\ell+m+n)$. We decompose the Lie algebra $\frak g$ of $G$ and the tangent
space $\frak p$ of $G/K$, by using the generalized flag manifolds $G/H =
\SU(\ell+m+n)/\s(\U(\ell)\times\U(m)\times\U(n))$.
  We parametrize scalar products on the 2-dimensional center of the Lie algebra
of $H$, and we consider $G$-invariant and left invariant metrics determined by
$\Ad(\s(\U(\ell)\times\U(m)\times\U(n))$-invariant scalar products on $\frak g$
and $\frak p$ respectively. Then we compute their Ricci tensor for such
metrics.
  We prove existence of $\Ad(\s(\U(1)\times\U(2)\times\U(2))$-invariant
Einstein metrics on $V_3\bb{C}^{5}=\SU(5)/\SU(2)$,
$\Ad(\s(\U(2)\times\U(2)\times\U(2))$-invariant Einstein metrics on
$V_4\bb{C}^{6}=\SU(6)/\SU(2)$, and
$\Ad(\s(\U(m)\times\U(m)\times\U(n))$-invariant Einstein metrics on
$V_{2m}\bb{C}^{2m+n}=\SU(2m+n)/\SU(n)$. We also prove existence of
$\Ad(\s(\U(1)\times\U(2)\times\U(2))$-invariant Einstein metrics on the compact
Lie group $\SU(5)$, which are not naturally reductive. The Lie group $\SU(5)$
is the special unitary group of smallest rank known for the moment, admitting
non naturally reductive Einstein metrics. Finally, we show that the compact Lie
group $\SU(4+n)$ admits two non naturally reductive
$\Ad(\s(\U(2)\times\U(2)\times\U(n)))$-invariant Einstein metrics for $ 2 \leq
n \leq 25$, and four non naturally reductive Einstein metrics for $n\ge 26$.
This extends previous results of K.~ Mori about non naturally reductive
Einstein metrics on $\SU(4+n)$ ($n \geq 2$).
