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 $frac{1}{2}({rm dim} , G - {rm rank} , G)$.
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