We characterize three classes of demand systems, all of which are defined nonparametrically: homothetic demand systems with a single aggregator (HSA), those with direct implicit additivity (HDIA), and those with indirect implicit additivity (HIIA). For any number of goods, all the cross-price effects are summarized by one price aggregator in the HSA class and by two price aggregators in the HDIA and HIIA classes. Each of these three classes contains CES as a special case. Yet, they are pairwise disjoint with the sole exception of CES. Thus, these classes of homothetic demand systems offer us three alternative ways of departing from CES.