Let
$R$
be a prime ring of characteristic diòerent from
$2$
, let
${{Q}_{r}}$
be its right Martindale quotient ring, and let
$C$
be its extended centroid. Suppose that
$F$
is a generalized skew derivation of
$R,\,L$
a non-central Lie ideal of
$R,\,0\,\ne \,a\,\in \,R,\,m\,\ge \,0$
and
$n,\,s\,\ge \,1$
fixed integers. If
$$a{{\left( {{u}^{m}}F\left( u \right){{u}^{n}} \right)}^{s}}\,=\,0$$
for all
$u\,\in \,L$
, then either
$R\,\subseteq \,{{M}_{2}}\left( C \right)$
, the ring of
$2\,\times \,2$
matrices over
$C$
, or
$m\,=\,0$
and there exists
$b\,\in \,{{Q}_{r}}$
such that
$F\left( x \right)\,=\,bx$
, for any
$x\,\in \,R$
, with
$ab\,=\,0$
.