  This paper studies the local and global aspects of semi-toric integrable
systems, introduced by Vu Ngoc, using ideas stemming from the theory of
Hamiltonian S^1-spaces developed by Karshon. First, we show how any labeled
convex polygon associated to a semi-toric system (as defined by Vu Ngoc)
determines Karshon's labeled directed graph which classifies the underlying
Hamiltonian S^1-space up to isomorphism. Then we characterize adaptable
semi-toric systems, i.e. those whose underlying Hamiltonian S^1-action can be
extended to an effective Hamiltonian T^2-action, as those which have at least
one associated convex polygon which satisfies the Delzant condition.
