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We introduce a model Hamiltonian that describes, for different choices of the parameters, simple anharmonic models for a solid. We have applied the Painleve test to identify integrable and non-integrable cases. In the integrable cases the identification has been confirmed by deriving explicit expressions for the additional conserved quantities. The analysis demonstrates the sensitivity of lattice integrability to both the order of the anharmonicity and the nature of the boundary conditions.

Integrability of Low Particle-number Models for Solids