  Carter constant is a non-trivial conserved quantity of motion of a particle
moving in stationary axisymmetric spacetime. In the version of the theorem
originally given by Carter, due to the presence of two Killing vectors, the
system effectively has two degrees of freedom. We propose an extension to the
first version of Carter's theorem to a system having three degrees of freedom
to find two functionally independent Carter-like integrals of motion. We
further generalize the theorem to a dynamical system with $N$ degrees of
freedom. We further study the implications of Carter Constant to
Superintegrability and present a different approach to probe a Superintegrable
system. Our formalism gives another viewpoint to a Superintegrable system using
the simple observation of separable Hamiltonian according to Carter's criteria.
We then give some examples by constructing some 2-Dimensional superintegrable
systems based on this idea and also show that all 3-D simple classical
Superintegrable potentials are also Carter separable.
