Recent studies on spin transport in integrable quantum and classical spin chains have shown a superdiffusive behaviour that is well-known in the Kardar-Parisi-Zhang (KPZ) universality class. Theoretical developments have highlighted the role of integrability and spin-symmetry for the KPZ superdiffusion. Nevertheless, it is a challenging task to understand their precise roles. In this work, we consider a classical integrable spin chain which possesses analogy with the quantum Heisenberg spin-1/2 model. Our numerical study shows that the KPZ behaviour remains stable when one considers integrability-breaking but spin-symmetry preserving terms, even in the non-perturbative regime. Energy correlations, however, exhibit diffusive behaviour there. Also, we check the classical analog of out-of-time-ordered correlator (OTOC) and Lyapunov exponents to establish the presence of chaos for the integrability-broken cases. (This is based on arXiv:2205.03858.)
Moreover, we study spin and energy transport under nonequilibrium conditions. We recover anomalous behaviour for spin and diffusive behaviour for energy in the non-perturbative regime.