The one-dimensional Holstein model and Holstein-Hubbard model have no nontrivial local conserved quantities other than the Hamiltonian and total fermion number.
Mazur, Non-ergodicity of phase functions in certain systems, Physica43, 533 (1969)
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Proof of the absence of local conserved quantities in the Holstein model
The one-dimensional Holstein model and Holstein-Hubbard model have no nontrivial local conserved quantities other than the Hamiltonian and total fermion number.