  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 $L\leq 32$ 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.
