  Recent investigations have observed superdiffusion in integrable classical
and quantum spin chains. An intriguing connection between these spin chains and
Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical
developments (e.g. generalized hydrodynamics) have highlighted the role of
integrability as well as spin-symmetry in KPZ behaviour. However understanding
their precise role on superdiffusive transport still remains a challenging
task. The widely used quantum spin chain platform comes with severe numerical
limitations. To circumvent this barrier, we focus on a classical integrable
spin chain which was shown to have deep analogy with the quantum
spin-$\frac{1}{2}$ Heisenberg chain. Remarkably, we find that KPZ behaviour
prevails even when one considers integrability-breaking but spin-symmetry
preserving terms, strongly indicating that spin-symmetry plays a central role
even in the non-perturbative regime. On the other hand, in the non-perturbative
regime, we find that energy correlations exhibit clear diffusive behaviour. We
also study the classical analog of out-of-time-ordered correlator (OTOC) and
Lyapunov exponents. We find significant presence of chaos for the
integrability-broken cases even though KPZ behaviour remains robust. The
robustness of KPZ behaviour is demonstrated for a wide class of spin-symmetry
preserving integrability-breaking terms.
