On homogeneous hypersurfaces in $${mathbb {C}}^3$$

We consider a family (M_t^n), with (ngeqslant 2), (t>1), of real hypersurfaces in a complex affine n-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 embeddability of (M_t^n) in  ({mathbb {C}}^n) for (n=3,7). In our earlier article we showed that (M_t^7) is not embeddable in  ({mathbb {C}}^7) for every t and that (M_t^3) is embeddable in  ({mathbb {C}}^3) for all (1. In the present paper, we improve on the latter result by showing that the embeddability of (M_t^3) in fact takes place for (1. This is achieved by analyzing the explicit totally real embedding of the sphere (S^3) in ({mathbb {C}}^3) constructed by Ahern and Rudin. For (tgeqslant {sqrt{(2+sqrt{2})/3}}), the problem of the embeddability of (M_t^3) remains open.