  We investigate dynamical fluctuations of transferred magnetization in the
one-dimensional lattice Landau--Lifshitz magnet with uniaxial anisotropy,
representing an emblematic model of interacting spins. We demonstrate that the
structure of fluctuations in thermal equilibrium depends radically on the
characteristic dynamical scale. In the ballistic regime, typical fluctuations
are found to follow a normal distribution and scaled cumulants are finite. In
stark contrast, on the diffusive and superdiffusive timescales, relevant
respectively for the easy-axis and isotropic magnet at vanishing total
magnetization, typical fluctuations are no longer Gaussian and, remarkably,
scaled cumulants are divergent. The observed anomalous features disappear upon
breaking integrability, suggesting that the absence of normal fluctuations is
intimately tied to the presence of soliton modes. In a nonequilibrium setting
of the isotropic magnet with weakly polarized step-profile initial state we
find a slow drift of dynamical exponent from the superdiffusive towards the
diffusive value.
