  In this expository paper, we provide an overview of the Gelfand-Zeiltin
integrable system on the Lie algebra of $n\times n$ complex matrices
$\fgl(n,\C)$ introduced by Kostant and Wallach in 2006. We discuss results
concerning the geometry of the set of strongly regular elements, which consists
of the points where Gelfand-Zeitlin flow is Lagrangian. We use the theory of
$K_{n}=GL(n-1,\C)\times GL(1,\C)$-orbits on the flag variety $\mathcal{B}_{n}$
of $GL(n,\C)$ to describe the strongly regular elements in the nilfiber of the
moment map of the system. We give an overview of the general theory of orbits
of a symmetric subgroup of a reductive algebraic group acting on its flag
variety, and illustrate how the general theory can be applied to understand the
specific example of $K_{n}$ and $GL(n,\C)$.
