  We study the space $X_h$ of Hermitian matrices having staircase form and the
given simple spectrum. There is a natural action of a compact torus on this
space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and
its smooth type is independent of the spectrum. Morse theory is then used to
show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly
formal manifold. The equivariant and ordinary cohomology of $X_h$ are described
using GKM-theory. The main goal of this paper is to show the connection between
the manifolds $X_h$ and the semisimple Hessenberg varieties well-known in
algebraic geometry. Both the spaces $X_h$ and Hessenberg varieties form
wonderful families of submanifolds in the complete flag variety. There is a
certain symmetry between these families which can be generalized to other
submanifolds of the flag variety.
