Twisted traces of quantum intertwiners and quantum dynamical R-matrices corresponding to generalized Belavin-Drinfeld triples

We consider weighted traces of products of intertwining operators for quan tum groups U-q(g), suitably twisted by a "generalized Belavin-Drinfeld triple". We derive two commuting sets of difference equations- the (twisted) Macdonald-Ruijsenaars system and the (twisted) quantum Knizhnik-Zamolodchikov-Bernard (qKZB) system. These systems involve the nonstandard quantum R-matrices defined in a previous joint work with T. Schedler ([ESS]). When the generalized Belavin-Drinfeld triple comes from an automorphism of the Lie algebra g, we also derive two additional sets of difference equations, the dual Macdonald-Ruijsenaars system and the dual qKZB equations.