Turbiner's Euclidean-geometry separation conjecture

Let $H$ be a Lie algebraic Schrödinger operator defined on a 2-dimensional manifold. The symbol of $H$ engenders a Euclidean geometry when the corresponding Gaussian curvature is zero. The spectral equation is

$$H\psi=E\psi.$$

Turbiner's second conjecture. If the symbol of $H$ engenders a Euclidean geometry, then the spectral equation $H\psi=E\psi$ can be solved by separation of variables. The paper states that this conjecture is false in general and provides a counterexample to the original conjecture.