  A real form $G_0$ of a complex semisimple Lie group $G$ has only finitely
many orbits in any given compact $G$-homogeneous projective algebraic manifold
$Z=G/Q$. A maximal compact subgroup $K_0$ of $G_0$ has special orbits $C$ which
are complex sub-manifolds in the open orbits of $G_0$. These are referred to as
cycles. The cycles intersect Shubert varieties $S$ transversely in finitely
many points. In particular, determining these points of intersection yields a
description of the topological class of the given cycle. This was carried out
for all real forms of $SL(n,\mathbb{C})$ in the work of A. Brecan. Our work
here is devoted to the real forms of the other classical groups,
$Sp(2n,\mathbb{C})$ and $SO(n,\mathbb{C})$. For the manifold $Z=G/B$ of
complete flags the points of intersection in $S\cap C$ are described, in
particular the number of such is computed. For certain real forms, e.g.,
$Sp(2n,\mathbb{R})$ and $SO^*(2n)$, remarkably simple formulas are proved. In
other cases the results are algorithmic in nature.
