Completely integrable systems always admit more alternative Hamiltonian descriptions. The geometrical formulation of quantum systems shows that similar conclusions hold true also for quantum systems. In addition, the description of quantum systems on Hilbert manifolds, e.g., the complex projective space, shows that not only quantum systems admit alternative Hamiltonian descriptions, they also admit alternative linear descriptions.
Giuseppe Marmo. Giovanni Sparano. Gaetano Vilasi. "Classical and Quantum Symmetries Reduction and Integrability Problem." J. Geom. Symmetry Phys. 31 105 - 117, 2013. https://doi.org/10.7546/jgsp-31-2013-105-117