an integrable system
An integrable system is a rare, beautiful special case: a mechanical system regular enough to be solved exactly, forever, with no chaos. Its motion is orderly and quasi-periodic -- the phase-space trajectory winds smoothly around a doughnut-shaped surface rather than wandering wildly. Most of the exactly solvable problems you ever meet (the harmonic oscillator, the Kepler problem, the free rigid body) are integrable, and this is precisely why they can be solved.
The precise criterion is Liouville integrability: a system with n degrees of freedom is integrable if it possesses n independent conserved quantities F_1 = H, F_2, ..., F_n that are in involution, meaning all their Poisson brackets vanish, {F_i, F_j} = 0. The Liouville-Arnold theorem then guarantees that the bounded motion lies on n-dimensional invariant tori in the 2n-dimensional phase space, that action-angle variables exist, and that the motion is a linear winding around each torus with fixed frequencies. Having as many independent commuting conserved quantities as degrees of freedom is exactly what 'enough constraints to solve it' means.
Integrability matters because it is the exception, not the rule, and knowing where it holds organizes all of mechanics. Genuinely integrable systems are non-generic; almost any perturbation destroys exact integrability, and the KAM theorem describes how the invariant tori partly survive and partly dissolve into chaos. Integrability also reaches far beyond classical mechanics -- into solitons, exactly solvable models in statistical mechanics, and quantum integrable systems -- where the same idea (a full set of commuting conserved quantities) again tames a problem that would otherwise be hopeless.
The two-body Kepler problem (one planet around the Sun) is integrable: energy, total angular momentum, and the direction of the Laplace-Runge-Lenz vector are all conserved and in involution, giving closed elliptical orbits that never precess. Add a second planet and those extra conservation laws are gone -- the three-body problem is non-integrable and generically chaotic.
Kepler's two-body problem is integrable (closed ellipses); the three-body problem is not, which is why planetary orbits precess and can be chaotic.
Conserved quantities must be in involution (mutually Poisson-commuting), not merely numerous. A system can have several conserved quantities and still be non-integrable if they do not all commute. 'Integrable' is a strong, structural property -- and generic Hamiltonian systems do not have it.