  In the present Letter we present an analytical and numerical solution of the
self-consistent mode-coupling equations for the problem of heat conductivity in
one-dimensional systems. Such a solution leads us to propose a different
scenario to accomodate the known results obtained so far for this problem. More
precisely, we conjecture that the universality class is determined by the
leading order of the nonlinear interaction potential. Moreover, our analysis
allows us determining the memory kernel, whose expression puts on a more firm
basis the previously conjectured connection between anomalous heat conductivity
and anomalous diffusion.
