This paper describes a novel approach to handling translation divergences in a Generation-Heavy Hybrid Machine Translation (GHMT) system. The translation divergence problem is usually reserved for Transfer and Interlingual MT because it requires a large combination of complex lexical and structural mappings. A major requirement of these approaches is the accessibility of large amounts of explicit symmetric knowledge for both source and target languages. This limitation renders Transfer and Interlingual approaches ineffective in the face of structurally-divergent language pairs with asymmetric resources. GHMT addresses the more common form of this problem, source-poor/targetrich, by fully exploiting symbolic and statistical target-language resources. This non-interlingual non-transfer approach is accomplished by using target-language lexical semantics, categorial variations and subcategorization frames to overgenerate multiple lexico-structural variations from a target-glossed syntactic dependency of the source-language sentence. The symbolic overgeneration, which accounts for different possible translation divergences, is constrained by a statistical target-language model.