In recent papers


<https://arxiv.org/abs/2101.05520>


<https://arxiv.org/abs/2001.06911>


(super)integrable systems on quiver varieties for cyclic and comet-shaped quivers are constructed.


My question: are there heuristics and/or conjectures about the existence of (super)integrable systems on more general quiver varieties (i.e. quiver varieties associated to more general quivers)?


Note: in <https://arxiv.org/abs/2001.06911> the authors state that "It is generally expected that Nakajima quiver varieties ought to be algebraically completely integrable Hamiltonian systems with a Hitchin-like fibration" but I am wondering what is the status of such expectations and how precisely they have been formulated. In <https://arxiv.org/abs/2001.06911> the authors also refer to Nakajima's paper "Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras" for a discussion of this question but I could not find such a discussion there in a direct form, at least there is no mentioning of the words 'integrable system' there.



(Super)integrable systems on quiver varieties
<http://arxiv.org/abs/2001.06911> referred my paper. But I meant more formal analogy between quiver varieties and Hitchin moduli spaces, such as hyper-Kaehler structure, S^1 action scaling the symplectic form, etc. I also pointed out that an analog of the Hitchin integrable system is the `affinization' morphism. It is often resolution of singularities, and generic fibers are points. Hence it is not an integrable system.


A comet-shaped quiver is `skelton' of punctured Riemann surface, and its quiver variety is additive analog of the Hitchin moduli space. This analogy goes back (at least) to <https://arxiv.org/abs/math/0103101>. Therefore it is natural to look for an integrable system in this case. (I do not mean that <http://arxiv.org/abs/2001.06911> was known before.)


Another example is a quiver variety of affine type A. It is known to be isomorphic to Coulomb branch of a quiver gauge theory of affine type A, different from the original in general. (<https://arxiv.org/abs/1606.02002>) By a general property of the Coulomb branch, it is equipped with an integrable system. <https://arxiv.org/abs/1601.03586>.


