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We study structures of solutions for SUSY Minkowski F-term equations on twotoroidal orientifolds with $h^{2, 1} = 1$. Following our previous study\cite{Ishiguro:2020tmo}, with fixed upper bounds of a flux D3-brane charge$N_{\rm flux}$, we obtain a whole Landscape and a distribution of degeneraciesof physically-distinct solutions for each case. In contrast to our previousstudy, we consider a non-factorizable toroidal orientifold and its Landscape onwhich $SL(2, \mathbb{Z})$ is violated into a certain congruence subgroup, as ithad been known in past studies. We find that it is not the entire duality groupthat a complex-structure modulus $U$ enjoys but its outer semi-direct productwith a "scaling" outer automorphism group. The fundamental region is enlargedto include the $|U| < 1$ region. In addition, we find that high degeneracy isobserved at an elliptic point, not of $SL(2, Z)$ but of the outer automorphismgroup. Furthermore, $\mathbb{Z}_2$-enhanced symmetry is realized on theelliptic point. The outer automorphism group is exceptional in the sense thatit is consistent with a symplectic basis transformation of backgroundthree-cycles, as opposed to the outer automorphism group of $SL(2,\mathbb{Z})$. We also compare this result with Landscape of anotherfactorizable toroidal orientifold.

Flux Landscape with Enhanced Symmetry Not on $SL(2, \mathbb{Z})$ Elliptic Points