s they contain both the $Q_L$ and $Q_R$ states in their spectra, whereas vacua with only symmetric boundary conditions with respect to the set $\{y,\omega\vert{\overline y},{\overline\omega}\}^{1,\cdots,6}$ do not contain $Q_R$ states and are therefore categorically excluded. It is of further interest to note that in the case of the LRS models the chirality of the $Q_L+L_L$ and $Q_R+L_R$ is similarly affected [@lrs]. However, there it is compensated by the chirality of the ${\overline\eta}^j$ worldsheet fermions leading to opposite charges under the $U(1)_j$ gauge symmetries. The SLM models [@slm] are obtained by combining the PS and FSU5 breaking vectors. Therefore, the SLM models produce complete ${\bf16}$ multiplets decomposed under the SLM group and with equal $U(1)_j$ charges. The SU421 class of models is the only case that is excluded in vacua with symmetric internal boundary conditions.

Conclusion
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In this paper we discussed the classification of the SU421 models with symmetric internal boundary conditions. This continues the development of the classification program initiated in ref. [@fknr], which led to the discovery of spinor–vector duality [@spinvecdual] and exophobic string vacua [@acfkr; @cfr; @SU6SU2]. The novel feature in the classification of the SU421 models compared to the PS and FSU5 vacua is the introduction of two basis vectors that break the $SO(10)$ symmetry. An appealing feature of the SU421 models is the admission of both the triplet–doublet as well as the doublet–doublet splitting mechanism, which is shared only with the standard–like models. However, as we showed in section \[nonv\] these models cannot accommodate the weak $SU(2)$ singlet states of the Standard Model and are therefore excluded. The next step in our classification program is the classification of standard–like models that will be reported in a future publication.

Acknowledgements
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We are grateful to John Rizos for fruitful discussions. The work of A.F. is partially supported by th