  Locally any completely integrable system is maximally superintegrable system
such as we have the necessary number of the action-angle variables. The main
problem is the construction of the single-valued additional integrals of motion
on the whole phase space by using these multi-valued action-angle variables.
Some constructions of the additional integrals of motion for the St\"ackel
systems and for the integrable systems related with two different quadratic
$r$-matrix algebras are discussed. Among these system there are the open
Heisenberg magnet and the open Toda lattices associated with the different root
systems.
