Bifurcation diagram and the discriminant of a spectral curve of integrable systems on Lie algebras

A bifurcation diagram is a stratified (in general, nonclosed) set and is one of the efficient tools of studying the topology of the Liouville foliation. In the framework of the present paper, the coincidence of the closure of a bifurcation diagram $\overline\Sigma$ of the moment map defined by functions obtained by the method of argument shift with the closure of the discriminant $\overline D_z$ of a spectral curve is proved for the Lie algebras $\operatorname{sl}(n+1)$, $\operatorname{sp}(2n)$, $\operatorname{so}(2n+1)$, and $\operatorname{g}_2$. Moreover, it is proved that these sets are distinct for the Lie algebra $\operatorname{so}(2n)$.
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