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