  Let $\mathcal{G}$ be the projective plane curve defined over $\mathbb{F}_q$
given by $$aX^nY^n-X^nZ^n-Y^nZ^n+bZ^{2n}=0,$$ where $ab\notin\{0,1\}$, and for
each $s\in\{2,\ldots,n-1\}$, let $\mathcal{D}_s^{P_1,P_2}$ be the
base-point-free linear series cut out on $\mathcal{G}$ by the linear system of
all curves of degree $s$ passing through the singular points $P_1=(1:0:0)$ and
$P_2=(0:1:0)$ of $\mathcal{G}$. The present work determines an upper bound for
the number $N_q(\mathcal{G})$ of $\mathbb{F}_q$-rational points on the
nonsingular model of $\mathcal{G}$ in cases where $\mathcal{D}_s^{P_1,P_2}$ is
$\mathbb{F}_q$-Frobenius classical. As a consequence, when $\mathbb{F}_q$ is a
prime field, the bound obtained for $N_q(\mathcal{G})$ improves in several
cases the known bounds for the number $n_P$ of chords of an affinely regular
polygon inscribed in a hyperbola passing through a given point $P$ distinct
from its vertices.
