  The $N\pi^0\pi^0$ decays of positive-parity $N^*$ and $\Delta^*$ resonances
at about 2\,GeV are studied at ELSA by photoproduction of two neutral pions off
protons. The data reveal clear evidence for several intermediate resonances:
$\Delta(1232)$, $N(1520){3/2^-}$, and $N(1680){5/2^+}$, with spin-parities
$J^P=3/2^+$, $3/2^-$, and $5/2^+$. The partial wave analysis (within the
Bonn-Gatchina approach) identifies $N(1440)1/2^+$ and the $N(\pi\pi)_{\rm
S-wave}$ (abbreviated as $N\sigma$ here) as further isobars, and assigns the
final states to the formation of nucleon and $\Delta$ resonances and to
non-resonant contributions. We observe the known $\Delta(1232)\pi$ decays of
$\Delta(1910)1/2^+$, $\Delta(1920)3/2^+$, $\Delta(1905)5/2^+$,
$\Delta(1950)7/2^+$, and of the corresponding spin-parity series in the nucleon
sector, $N(1880)1/2^+$, $N(1900)3/2^+$, $N(2000)5/2^+$, and $N(1990)7/2^+$. For
the nucleon resonances, these decay modes are reported here for the first time.
Further new decay modes proceed via $N(1440)1/2^+\pi$, $N(1520)3/2^-\pi$,
$N(1680)5/2^+\pi$, and $N\sigma$. The latter decay modes are observed in the
decay of $N^*$ resonances and at most weakly in $\Delta^*$ decays. It is argued
that these decay modes provide evidence for a 3-quark nature of $N^*$
resonances rather than a quark-diquark structure.
