Relation between solutions to Yang-Baxter equations, integrability and exact solvability?

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Wikipedia mentions that there is an implication: Yang-Baxter solutions yield integrable models, what 1D systems concerns.

In arbitrary dimensions, what is the relation, if any, between solutions to Yang-Baxter equations, integrability and exact solvability?

If somebody could provide a no-go theorem or cases, where integrability was already ruled out, it would be great.

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user c.p.

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Very roughly, the presence of an R-matrix (a solution of the YBE) provides a mechanism for constructing sufficiently many (infinitely many for field-theoretic models) integrals of motion, and this ensures integrability.

As for the YBE in statistical physics, there is a classical book

R.J. Baxter, Exactly solved models in statistical mechanics.

See e.g. also the references at http://www.encyclopediaofmath.org/index.php/Yang–Baxter_equation
answered Jun 15, 2015 by (95 points)

The question was primarily about the higher-dimensional case. Do you have something to add there?

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