Integrable systems and Poisson-Lie T-duality: A finite dimensional example

We study the deep connection between integrable models and Poisson-Lie T-duality working on a finite dimensional example constructed on SL(2,C) and its Iwasawa factors SU(2) and B. We shown the way in which the Adler-Kostant-Symes theory and collective dynamics combine to solve the equivalent systems by solving the factorization problem of an exponential curve in SL(2,C). It is shown that the Toda system embraces the dynamics of the systems on SU(2) and B.