  We consider a class of two-dimensional Schr\"odinger operator with a singular
interaction of the $\delta$ type and a fixed strength $\beta$ supported by an
infinite family of concentric, equidistantly spaced circles, and discuss what
happens below the essential spectrum when the system is amended by an
Aharonov-Bohm flux $\alpha\in [0,\frac12]$ in the center. It is shown that if
$\beta\ne 0$, there is a critical value $\alpha_\mathrm{crit} \in(0,\frac12)$
such that the discrete spectrum has an accumulation point when
$\alpha<\alpha_\mathrm{crit} $, while for $\alpha\ge\alpha_\mathrm{crit} $ the
number of eigenvalues is at most finite, in particular, the discrete spectrum
is empty for any fixed $\alpha\in (0,\frac12)$ and $|\beta|$ small enough.
