  Let $p$ be an odd prime, $F/{\Bbb Q}$ an abelian totally real number field,
$F_\infty/F$ its cyclotomic ${\Bbb Z}_p$-extension, $G_\infty = Gal (F_\infty /
{\Bbb Q}),$ ${\Bbb A} = {\Bbb Z}_p [[G_\infty]].$
  We give an explicit description of the equivariant characteristic ideal of
$H^2_{Iw} (F_\infty, {\Bbb Z}_p(m))$ over ${\Bbb A}$ for all odd $m \in {\Bbb
Z}$ by applying M. Witte's formulation of an equivariant main conjecture (or
"limit theorem") due to Burns and Greither. This could shed some light on
Greenberg's conjecture on the vanishing of the $\lambda$-invariant of
$F_\infty/F.$
