We introduce a notion of weakly log-canonical Poisson structures on positive varieties with potentials. Such a Poisson structure is log-canonical up to terms dominated by the potential. To a compatible real form of a weakly log-canonical Poisson variety we assign an integrable system on the product of a certain real convex polyhedral cone (the tropicalization of the variety) and a compact torus.
We apply this theory to the dual Poisson-Lie group G∗ of a simply-connected semisimple complex Lie group G. We define a positive structure and potential on G∗ and show that the natural Poisson-Lie structure on G∗ is weakly log-canonical with respect to this positive structure and potential.
For K \subset G the compact real form, we show that the real form K^* \subset G^* is compatible and prove that the corresponding integrable system is defined on the product of the decorated string cone and the compact torus of dimension \frac{1}{2}({\rm dim} \, G - {\rm rank} \, G).
