You can consider Nekrasov-Shatashvili for application to Toda chain or Calagero-Moser type systems. That nominally looks like a string or high energy particle link, but those systems are integrable chain systems. Looking for any experiments on these sorts of systems, the setup might be some optical trap like http://www.nature.com/nature/journal/v440/n7086/full/nature04693.html

I don't know if you would call this sort of Atomic, Molecular, Optical system as a cheating answer because it is designed to have that integrable structure.

Same principal for any other $\mathcal{N}=2$ way of getting any other integrable chain that could be engineered with laser traps.