  We introduce an atlas adapted to the Toda flow on the manifold of full flags
of any non-compact real semisimple Lie algebra, and on its Hessenberg-type
submanifolds. In our local coordinates the Toda flow becomes linear. We use
these new coordinates to show that the Toda flow on the manifold of full flags
is Morse-Smale, which generalizes the main result of \cite{CSS1} to arbitrary
non-compact real semisimple Lie algebras. As a byproduct we describe new
features of classical constructions in matrix theory.
