We prove an all-or-nothing theorem for all isotropic nearest-neighbor spin chains with arbitrary spin $S$: the system is either completely integrable (it has all $k$-local conserved quantities) or nonintegrable (it has no nontrivial local conserved quantity). This result excludes the possibility of an intermediate system with some but not all local conserved quantities, which solves in the affirmative the Grabowski-Mathieu conjecture. In addition, these two cases are decided by whether a single, easily computable quantity $D^3$ is zero or not. This theorem also serves as a complete classification of integrability and nonintegrability for $S\le 13.5$, suggesting that all the integrable models are exhausted by the Yang-Baxter solvable families.

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• Quantum many-body systems
• Exact solutions for many-body systems
• Integrable systems
• Spin chains