  A K-contact manifold is a smooth manifold M with a contact form whose Reeb
flow preserves a Riemannian metric on M. Main examples are Sasakian manifolds.
Our results in this paper are four results i), ii), iii) and iv) below obtained
by the application of the Morse theory to the contact moment maps on closed
5-dimensional K-contact manifolds of rank 2. 5-dimensional K-contact manifolds
of rank 2 have the lowest dimension among K-contact manifolds with nontrivial
contact moment maps which are not toric. i) Correspondence between the
isomorphism classes of K-contact manifolds and graphs of isotropy data, ii)
Classification of K-contact manifolds up to contact blowing up and down, iii)
Every closed 5-dimensional K-contact manifold of rank 2 has a Sasakian metric,
iv) A sufficient condition for closed 5-dimensional K-contact manifolds of rank
2 to be contact toric.
