We investigate the high-temperature dynamical conductivity $sigma(omega)$ in two one-dimensional integrable quantum lattice models: the anisotropic XXZ spin chain and the Hubbard chain. The emphasis is on the metallic regime of both models, where besides the ballistic component, the regular part of conductivity might reveal a diffusive-like transport. To resolve the low-frequency dynamics, we upgrade the microcanonical Lanczos method enabling studies of finite-size systems with up to $Lleq 32$ sites for XXZ spin model with the frequency resolution $delta omega sim 10^{-3} J$. Results for the XXZ chain reveal a fine structure of $sigma(omega)$ spectra, which originates from the discontinuous variation of the stiffness, previously found at commensurate values of the anisotropy parameter $Delta$. Still, we do not find a clear evidence for a diffusive component, at least not for commensurate values of $Delta$, particularly for $Delta =0.5$, as well as for $Delta to 0$. Similar is the conclusion for the Hubbard model away from half-filling, where the spectra reveal more universal behavior.
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