One-dimensional systems often display anomalous energy diffusion and transport. Besides their intrinsic theoretical interest, these features are relevant for low-dimensional structures in view of the applications for nanoscale heat transfer. After a short review of numerical, theoretical and experimental results on nonlinear systems, we discuss a model consisting of a chain of harmonic oscillators subject to conservative noise. The stationary equations for the covariance matrix under a temperature gradient can be exactly solved in the thermodynamic limit. In particular, we derive an analytical expression for the temperature profile, and the energy current. In this framework, an explicit representation of the nonequilibrium invariant measure can be provided in terms of the principal components analysis.