A **superintegrable Hamiltonian system** is a special class of Hamiltonian dynamical systems that possesses more integrals of motion than degrees of freedom. In classical mechanics, a Hamiltonian system is typically described by its Hamiltonian function, which encodes the total energy of the system. The system's behavior is determined by Hamilton's equations, which govern the time evolution of the system's phase space.
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A superintegrable Hamiltonian system with degrees of freedom has more than functionally independent integrals of motion, including an integrable set of in involution. At a regular nonzero Hamiltonian vector field, at most independent integral differentials can annihilate the flow. A maximally superintegrable system attains that bound. The three-dimensional Kepler orbit has five independent integrals through energy, angular momentum and the Laplace-Runge-Lenz vector. A closed orbit on one isolated resonance is weaker than maximal superintegrability on an open phase-space region.