The Heisenberg Hamiltonian is a highly symmetric Hamiltonian where group theory is very effective for the qualitative as well as the quantitative study of its eigenstates. For this matter, a system of nested algebras related to the Irreps of the symmetry group of the Hamiltonian was developed. Recently, experimental facts on magnetic materials showed that interactions lowering the symmetry, like the Dzialoshinski term, are necessary for their interpretation. A perturbation theory is developed for such Hamiltonians and applied to experimental data with very good agreement between theory and experiment.