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Brandhorst and Shimada described a large class of Enriques surfaces, called$(\tau,\overline{\tau})$-generic, for which they gave generators for theautomorphism groups and calculated the elliptic fibrations and the smoothrational curves up to automorphisms. In the present paper, we give lower boundsfor the non-degeneracy invariant of such Enriques surfaces, we show that inmost cases the invariant has generic value $10$, and we present the first knownexample of complex Enriques surface with infinite automorphism group andnon-degeneracy invariant not equal to $10$.

The non-degeneracy invariant of Brandhorst and Shimada families of Enriques surfaces