  We consider theories which explain the flatness of the power spectrum of
scalar perturbations in the Universe by conformal invariance, such as conformal
rolling model and Galilean Genesis. We show that to the leading {\it
non-linear} order, perturbations in all models from this class behave in one
and the same way, at least if the energy density of the relevant fields is
small compared to the total energy density (spectator approximation). We then
turn to the intrinsic non-Gaussianities in these models (as opposed to
non-Gaussianities that may be generated during subsequent evolution). The
intrinsic bispectrum vanishes, so we perform the complete calculation of the
trispectrum and compare it with the trispecta of local forms in various limits.
The most peculiar feature of our trispectrum is a (fairly mild) singularity in
the limit where two momenta are equal in absolute value and opposite in
direction (folded limit). Generically, the intrinsic non-Gaussianity can be of
detectable size.
