On the classification by Morimoto and Nagano

We consider a family $M_t^3$, with $t>1$, of real hypersurfaces in a complex affine $3$-dimensional quadric arising in connection with the classification of homogeneous compact simply-connected real-analytic hypersurfaces in ${\mathbb C}^n$ due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the CR-embeddability of $M_t^3$ in ${\mathbb C}^3$. In our earlier article we showed that $M_t^3$ is CR-embeddable in ${\mathbb C}^3$ for all $1<t<\sqrt{(2+\sqrt{2})/3}$. In the present paper we prove that $M_t^3$ can be immersed in ${\mathbf C}^3$ for every $t>1$ by means of a polynomial map. In addition, one of the immersions that we construct helps simplify the proof of the above CR-embeddability theorem and extend it to the larger parameter range $1<t<\sqrt{5}/2$.