  We report how a doublet of the symmetric oppositely tilted bistable resonance
peaks in a microring resonator with quadratic nonlinearity set for generation
of the second harmonic can transform into a Kerr-like peak on one side of the
linear cavity resonance and into a closed loop structure disconnected from the
quasi-linear resonance on the other. Both types of the nonlinear resonances are
associated with the formation of the soliton combs for dispersion profiles of a
typical LiNbO$_3$ microring. We report bright quasi-solitons propagating on a
weakly modulated low intensity background when the group velocity dispersions
have the opposite signs for the fundamental and second harmonic. We also show
exponentially localized solitons when the dispersion signs are the same.
Finally, we demonstrate that the transition between these two types of soliton
states is associated with the closure of the forbidden gap in the spectrum of
quasi-linear waves.
