  The structure theory for the quadratic algebra generated by first and second
order constants of the motion for 2D second order superintegrable systems with
nondegenerate (3-parameter) and or 2-parameter potentials is well understood,
but the results for the strictly 1-parameter case have been incomplete. Here we
work out this structure theory and prove that the quadratic algebra generated
by first and second order constants of the motion for systems with 4 second
order constants of the motion must close at order three with the functional
relationship between the 4 generators of order four. We also show that every
1-parameter superintegrable system is St\"ackel equivalent to a system on a
constant curvature space.
