We extend the theory of ground states of classical Heisenberg spin systems previously published to the case where the interaction with an external magnetic field is described by a Zeeman term. The ground state problem for the Heisenberg-Zeeman Hamiltonian can be reduced first to the relative ground state problem, and, in a second step, to the absolute ground state problem for pure Heisenberg Hamiltonians depending on an additional Lagrange parameter. We distinguish between continuous and discontinuous reduction. Moreover, there are various general statements about Heisenberg-Zeeman systems that will be proven under most general assumptions. One topic is the connection between the minimal energy functions Emin for the Heisenberg energy and Hmin
Nearby in the stack
for the Heisenberg-Zeeman energy which turn out to be essentially mutual Legendre-Fenchel transforms. This generalization of the traditional Legendre transform is especially suited to cope with situations where the function
Emin
is not convex and consequently there is a magnetization jump at a critical field. Another topic is magnetization and the occurrence of threshold fields
Bthr
and saturation fields
Bsat
, where we provide a general formula for the latter. We suggest a distinction between ferromagnetic and anti-ferromagnetic systems based on the vanishing of
Bsat
for the former ones. Parabolic systems are defined in such a way that
Emin
and
Hmin
have a particularly simple form and studied in detail. For a large class of parabolic systems the relative ground states can be constructed from the absolute ground state by means of a so-called umbrella family. Finally we provide a counter-example of a parabolic system where this construction is not possible.