  Recently introduced time-dependent renormalized-natural-orbital theory
(TDRNOT) is based on the equations of motion for the so-called natural
orbitals, i.e., the eigenfunctions of the one-body reduced density matrix.
Exact TDRNOT can be formulated for any time-dependent two-electron system in
either spin configuration. In this paper, the method is tested against
high-order harmonic generation (HHG) and Fano profiles in absorption spectra
with the help of a numerically exactly solvable one-dimensional model He atom,
starting from the spin-singlet ground state. Such benchmarks are challenging
because Fano profiles originate from transitions involving autoionizing states,
and HHG is a strong-field phenomenon well beyond linear response. TDRNOT with
just one natural orbital per spin in the helium spin-singlet case is equivalent
to time-dependent Hartree-Fock or time-dependent density functional theory
(TDDFT) in exact exchange-only approximation. It is not unexpected that TDDFT
fails in reproducing Fano profiles due to the lack of doubly excited,
autoionizing states. HHG spectra, on the other hand, are widely believed to be
well-captured by TDDFT. However, HHG spectra of helium may display a second
plateau that originates from simultaneous HHG in He$^+$ and neutral He. It is
found that already TDRNOT with two natural orbitals per spin is sufficient to
capture this effect as well as the Fano profiles on a qualitative level. With
more natural orbitals (6--8 per spin) quantitative agreement can be reached.
Errors due to the truncation to a finite number of orbitals are identified.
