It has been theoretically demonstrated that two spins (qubits or qutrits), coupled by exchange interaction only, undergo a coupling-based joint Landau-Majorana-Stückelberg-Zener (LMSZ) transition when a linear ramp acts on one of the two spins. Such a transition, under appropriate conditions on the parameters, drives the two-spin system toward a maximally entangled state. In this paper, effects on the quantum dynamics of the two qudits, stemming from the Dzyaloshinskii-Moriya (DM) and dipole-dipole (d-d) interactions, are investigated qualitatively and quantitatively. The enriched Hamiltonian model of the two spins shares with the previous microscopic one the same C2 symmetry which once more brings about an exact treatment of the new quantum dynamical problem. This paper transparently reveals that the DM and d-d interactions generate independent, enhancing or hindering, modifications in the dynamical behavior predicted for the two spins coupled exclusively by the exchange interaction. It is worthwhile to notice that, on the basis of the theory here developed, the measurement of the time evolution of the magnetization in a controlled LMSZ scenario can furnish information on the relative weights of the three kinds of couplings describing the spin system. This possibility is very important since it allows us in principle to legitimate the choice of the microscopic model to be adopted in a given physical scenario.
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