Quantum Monodromy problem for effective polyad Breaking Hamiltonian. The idea of quantum monodromy was recently developed, shown to be an important qualitative feature of many different realistic models. Starting from these examples new qualitative features of molecular systems leading naturally to generalized monodromy notions is used. Going finally to really extreme complex systems the relationship or link between phyllotaxis and monodromy is also possible. Recent proof of the persistence of monodromy under small perturbations spoiling the integrability strengthens the interest of applications to non-integrable systems.