  Recent studies have found that fluctuations of magnetization transfer in
integrable spin chains violate the central limit property. Here we revisit the
problem of anomalous counting statistics in the Landau-Lifshitz field theory by
specializing to two distinct anomalous regimes featuring a dynamical critical
point. By performing optimized numerical simulations using an integrable
space-time discretization we extract the algebraic growth exponents of
time-dependent cumulants which attain their threshold values. The distinctly
non-Gaussian statistics of magnetization transfer in the easy-axis regime is
found to converge towards the universal distribution of charged single-file
systems. At the isotropic point we infer a weakly non-Gaussian distribution,
corroborating the view that superdiffusive spin transport in integrable spin
chains does not belong to any known dynamical universality class.
