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 trip le". We derive two commuting sets of difference equations- the (twisted) Ma cdonald-Ruijsenaars system and the (twisted) quantum Knizhnik-Zamolodchikov -Bernard (qKZB) system. These systems involve the nonstandard quantum R-mat rices defined in a previous joint work with T. Schedler ([ESS]). When the g eneralized Belavin-Drinfeld triple comes from an automorphism of the Lie al gebra g, we also derive two additional sets of difference equations, the du al Macdonald-Ruijsenaars system and the dual qKZB equations.