Classical spin clusters: Integrability and dynamical properties

A pair of exchange‐coupled classical spins with biaxial exchange and single‐site anisotropy represents a Hamiltonian system with two degrees of freedom for which the integrability question is nontrivial. We have found that such a system is completely integrable if the model parameters satisfy a certain condition. For the integrable cases, the second integral of the motion (in addition to the Hamiltonian), which guarantees integrability, is determined explicitly. It can be reconstructed numerically by means of time averages of dynamical variables over all trajectories. In the nonintegrable cases, the existence of the time averages is still guaranteed, but they no longer define an analytic invariant, and their determination is subject to long‐time anomalies. Our numerical calculation of time averages for two lines of initial conditions reveals a number of interesting features of such nonanalytic invariants.