Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example is given based on the trigonometric Olshanetsky-Perelomov potential corresponding to the $A_2$ root system.

MSC Classifications:

35Q40, 53C30, 81R05 show english descriptions PDEs in connection with quantum mechanics
Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]
Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70] 35Q40 - PDEs in connection with quantum mechanics
53C30 - Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]
81R05 - Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70]

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